Fair Coalitions for Power-Aware Routing in Wireless Networks

Fir Colitions for Power-Awre Routing in Wireless Networks Rtul K. Guh, Crl A. Gunter nd Sswti Srkr University of Pennsylvni {rguh@ses, gunter@.cis, swti@.ee }.upenn.edu Astrct Severl power wre routing schemes hve een developed under the ssumption tht nodes re willing to scrifice their power reserves in the interest of the network s whole. But, in severl pplictions of prcticl utility, nodes re orgnized in groups, nd s result node is willing to scrifice in the interest of other nodes in its group ut not necessrily for nodes outside its group. Such groups rise nturlly s sets of nodes ssocited with single owner or tsk. We consider the premise tht groups will shre resources with other groups only if ech group experiences reduction in power consumption. When this is the cse the groups my form colition in which they route ech other s pckets. We demonstrte tht shring etween groups hs different properties from shring etween individuls nd investigte fir mutully-eneficil shring etween groups. In prticulr, we propose preto-efficient condition for group shring sed on mx-min firness clled fir colition routing. We propose distriuted lgorithms for computing the fir colition routing. Using these lgorithms we demonstrte tht fir colition routing llows different groups to mutully enefecilly shre their resources. I. INTRODUCTION Wireless networks typiclly consist of nodes tht must dischrge incresingly complex computing nd communiction functionlities despite constrints on power, ndwidth, size nd memory. Significnt progress hs een mde to improve hrdwre to ddress these needs nd much is eing done to develop softwre tht uses techniques like poweroptimizing lgorithms. Comprtively less hs een done to exploit shring mongst nodes s wy to ddress these chllenges. This is unfortunte, since shring cn yield gret enefits. A vriety of chllenges impede progress: () determining which resources cn e shred, () deciding when to shre resources, s shring would evidently involve cost, (c) deciding with whom to shre resources, nd (d) determining how to shre resources. Oftentimes, groups of nodes rther thn individul nodes re sic entities in the shring mechnism. The resource expenditure of the group s whole is more importnt thn tht of single node or the entire network. Groups re often formed on the sis of memership in n orgniztion or shred tsk. For exmple, employees of n orgniztion A my crry computers tht elong to A. When these devices form n d hoc network, they my shre resources with other devices with the ojective of minimizing the totl resource consumed y the devices in A, rther thn tht of ll devices in the network. Thus, the devices elonging to n orgniztion form nturl group. Werle computers involved in one distriuted computtion my form group. In sensor network, different groups would consist of Appered in: IEEE Conference on Decision nd Control (CDC ), Prdise Islnd, Bhms, Decemer. The reserch ws supported in prt y NSF EIA-888, NSF ANI-698, ONR N---75 nd NCR-383. sensors tht mesure different ttriutes such s temperture, pressure etc. In oth the ove cses, the resource consumed y groups is more importnt thn tht consumed y individul nodes s the distriuted computtion cn e performed nd the ttriutes cn e mesured even when some memers fil. The reserch in this cse must investigte issues pertinent to shring of resources from the perspective of groups. A group is n intermingled set of nodes hving purpose in common. We do not consider the motivtion ehind the group formtion, ut investigte the shring of resources mong different groups. The criticl resource we focus on is power. Nodes in wireless networks re powered y ttery, nd size limittions compel the usge of low lifetime tteries. This clls for judicious consumption of power. Normlly, communiction consumes higher power thn other opertions. Nodes shre power y routing ech others pckets, nd it is well-known tht multihop routing sustntilly decreses the overll power consumption of the network [5]. We ddress the reserch chllenges tht rise when nodes decide to route ech others pckets with the sole ojective of reducing the power consumption of their groups. We first enumerte these chllenges. The nodes in group shre power y routing ech other s pckets to common destintions. Groups re sid to form colitions when they route ech other s pckets. The first chllenge is to determine which groups would form colitions. Presumly, precondition for forming colitions mong groups is tht ech group communictes the sme mount of informtion to the chosen destintions while consuming less power fter the colition is formed. Whether or not the precondition is stisfied depends on the routing in the colition, nd the numer of possile routes cn e n exponentil function of the numer of nodes in the groups. There need not even exist routing tht reduces the power consumption of ech group. Fig.() nd () show tht if ech group consists of single node, then groups do not mutully enefit from the colition; ut this no longer holds if the groups consist of two or more nodes (Fig.(c)). The chllenge then is to nswer whether there exists t lest one joint routing tht mkes the colition mutully eneficil. The next chllenge is to decide the joint routing when the colition is formed. We will show in Section III-C tht the routing tht minimizes the totl power consumption of ll groups is not the right choice, s it my increse the power consumption of some groups despite minimizing the power consumption of the network s whole. The enefit of group due to the colition opertion is the decrese in its power consumption fter it joins the colition. We need to determine routing tht shres the enefit equitly. A simplistic pproch is to insist tht the groups ech get the sme enefit, ut this cn e wsteful if one group cn

() AP () Fig.. In () nd (), we show two different routings where node constitutes group A nd node constitutes group B. Both groups need to send trffic to the ccess point(ap). In () the frther node routes its trffic to nd sends to AP. So the routing in (), reduces the power cost of ut increses tht for. In () ech node routes directly to AP nd there is no reduction in power costs for oth groups. In (c) nodes nd constitute group A nd constitutes group B. Here cn send its trffic through nd cn in turn send through. This could result in decrese in the totl power for group A nd B s ginst the cse when the groups route to AP independently. gin enefit without hrming the others. A mx-min fir [] routing uses the following strtegy for pir of groups: determine the gretest minimum enefit to e gined y either of the two groups when shring nd mximize the enefit of the other group so long s the chnges do not reduce this minimum. This strtegy cn e generlized to multiple groups. The chllenge now is to compute mxmin fir power wre colition routing. We survey the relevnt literture in Section II. We provide mthemticl frmework for colition of two groups in Section III. This section presents severl interesting properties of colition routings. For exmple, mx-min fir power wre colition routing exhiits importnt chrcteristics tht do not hold for mx-min fir lloction of other resources such s ndwidth. We show tht the mxmin fir colition routing is gurnteed to ttin the desired minimum enefits for ech group should the colition e fesile. We present polynomil complexity lgorithm for computing the fir colition routing in Section IV. This lgorithm needs solving liner progrm t centrl processor, which requires the knowledge of the glol topology. We present distriuted computing scheme which llows the routing to e computed vi simple itertive computtions nd messge exchnges t ech prticipting node in Section V. All proofs cn e found in the technicl report [9]. II. RELATED WORK The existing reserch on efficient utiliztion of power in wireless networks cn e clssified into the following rod ctegories. The first mximizes the lifetime of ny given node through optimum ttery dischrge strtegy [6], [7]. The second vries the trnsmission power levels of nodes so s to control the network topology s desired [8], [], [3]. The third reduces the power consumption y optimizing severl prmeters t the MAC lyer [], []. The lst mximizes the lifetime of the network y lncing the power consumption of different nodes [], [5]. Another prevlent pproch is to route in ccordnce with power sed metric rther thn distnce metric [5]. However the common feture of the existing reserch is tht the sic entity is node. The performnce of the network is either quntified in terms of the ggregte performnce of the nodes or tht of the ottleneck node. However, in our cse the sic entity is group rther thn single AP AP (c) node, nd the opertions re colitions. The performnce ojective we consider is firness nd the issues significntly differ on ccount of the choice of the sic entity. We re concerned out the performnce of ech group rther thn the network s whole. Relying nd cching strtegies hve een proposed [], [8] for node coopertion where node decides to rely the requests of other nodes. The lgorithm in [] propels the network towrds preto optiml operting point. Our reserch is complementry in the sense tht we ssume tht group of nodes decide to route the pckets of other groups sed on the interest of the group s whole. We present n lgorithm tht otins specific preto optiml ojective, the mx-min fir operting point. III. MATHEMATICAL FRAMEWORK FOR COALITION OF GROUPS A. Power Model We first present the mthemticl model we use for power consumption [7], []. Let the trnsmitted energy per it e E t. Then the received energy t distnce d is E t d α where α is generlly etween to 6. The higher vlue of the exponent pplies for ostructed pths within uildings. We ssume tht the noise level is the sme t ll nodes. Let E r e the energy per it required to mintin threshold SNR t the receiving end. Then for successful communiction E t d α E r. The trnsmitted power then is of the form K E r Rd α where R is the it rte nd K = K E r is constnt. We will use α =which corresponds to the pthloss in closed res; however ll nlysis will hold for for ny α. B. Formultion For Single Group We consider network in which multiple nodes in group send trffic to n exit point (). This cn e motivted y severl commonplce pplictions. For exmple, consider wireless we-cfe, where users send pckets to common ccess point. In sensor networks mesurements must e communicted to exit nodes. In the first cse groups cn e formed on the sis of memership to different orgniztions while in the second, groups my e formed on the sis of tsks. We model the network nodes s Weighted Directed Grph G V,E,,W where V is the node set for the group, E is the edge set, is the exit point nd W denotes the edge weights R. R denotes the set of rel numers. Every node v V hs t lest one pth to node nd outdegree of is. The node set V nd the exit point re defined through their co-ordintes in the eucliden plne. The distnce d(v, v ) is the distnce etween node v V nd node v V {}. The distnce informtion cn e otined through power mesurements nd positioning lgorithms such s in []. Now we define the edge set E nd the corresponding weight set W. Let D denote the mximum distnce tht gurntees correct decoding of ny communiction etween two nodes. In other words D ensures n cceptle SNR level t the receiver. A directed edge exists from v V to v V {} if d(v, v ) <Dnd consequently (v, v ) E with weight w(v, v )=d(v, v ) nd w(v, v ) W. Note tht the exit point hs only incoming edges. Origin function O : V Rdefines the trffic originting t node v V. The grph G nd the origin functions re given.

Let the trffic on n edge (v, v ) e r(v, v ) R.If (v, v ) E then r(v, v )=. The totl outgoing trffic from node v is then v V {} r(v, v ) which is the lod on node v, L(v). The sum of the incoming trffic nd the originting trffic t node must equl the exiting trffic. Thus, v V r(v, v )=O(v)+ r(v,v)=l(v). () v V {} v V Trffic routing is n E dimensionl vector r whose components stisfy (). The components of r re the trffics on the corresponding edges. Given the routing, the power expenditure of node v, N r (v) is the power spent to trnsmit lod L(v) i.e., N r (v) =K r(v, v )d(v, v ) where v V {} K is the constnt s defined in Section III-A. The power expenditure of group P r is then the sum of the power expenditure over ll nodes of tht group i.e., P r = N r (v). The group optiml power expenditure v V P opt is the minimum vlue of P r over ll possile r. Here P opt corresponds to routing the trffic over the shortest pth from ny node v V to in terms of cost metric W. The shortest pth cn e otined through lgorithms like Dijkstr. Let v e the next hop node to v s otined from the shortest pth lgorithm. If N opt (v) is the power spent y node v under optiml routing, then N opt (v) =K L(v) d(v, v ) nd P opt = N opt (v). v V C. Colition of Groups We hve descried the terminology nd the equtions for group of nodes. Now consider two groups of nodes A nd B. Let their node sets e V nd V respectively nd optiml power expenditures efore forming colition e Popt nd Popt. Next we consider comined network with groups A nd B jointly routing to the exit point. The vertex set V for the comined network then is V V. The edge set E joint cn e found from V s follows. A directed edge exists from v V V to v V V {} if d(v, v ) <D nd consequently (v, v ) E joint with weight w(v, v )= d(v, v ). The origin functions for ll the nodes remin the sme. A colition routing in this network is vector whose components stisfy (). Note tht r(v, v )=if (v, v ) E joint. For n ritrry colition routing r, evlute the power expenditure for ech node. Let J r nd J r e the totl power expenditure for nodes in groups A nd B respectively, under routing r. J r = N r (v) nd J r = N r (v). v V v V Definition : Group enefit under colition routing r is the difference etween the power spent y the group under individul optiml routing efore merging, nd the power spent y the group for colition routing r. The group enefits under routing r form the enefit vector B r. Hence the enefit vector is B r (B r,b r ) with components B r = P opt J r nd B r = P opt J r. The ide ehind comining two groups is to reduce the totl power ech group ws spending initilly. Depending on Y (-,) x (-,) (,) Group A (,) (,.5) (-,) (,) Colition (,) Not to scle (,.5) (,) (-,) Group B Fig.. Groups A(, ) nd B(, ) route to the exit point. Ech node sends Mps the system, group colition my introduce some dditionl opertionl cost nd groups would wnt to enefit over nd ove this cost Definition : A colition is useful with routing r if min(b r,b r ) t. A colition is useful if there exists routing r such tht the colition is useful with routing r. We would present n lgorithm to compute such routing r if one exists. The choice of the threshold t, would depend on group policies nd the overhed for the colition. Definition 3: A Miniml colition routing is joint routing tht results in the optiml or the miniml totl power expenditure for groups A nd B comined. Next we illustrte the comintion of two groups with n exmple. Consider Fig. in which groups A nd B route to the exit point. Ech node genertes trffic t the rte of Mps. Optiml power expenditure for group A is + =nd for group B is +.5 9. For the miniml power colition routing shown, power expenditure forais +( ) =9nd for B is () +.5 3.6. for group A is 9=nd for B is 9 3.6 = 5. nd oth the components re positive. Consider now tht node hs higher lod to send, e.g., 5Mps. This will e relyed through in the colition routing of Fig.. Node will hve high power consumption() nd the enefit of group A will e negtive(-5). This illustrtes tht the miniml colition routing my not enefit ech group. Definition : A fesile enefit vector is one tht results from colition routing r tht stisfies (). The set of ll fesile enefit vectors is the fesile enefit region. D. Properties of the Fesile Region For the miniml colition routing, we cn find the power expenditure for ech node, i.e. N opt (v) for ech v (V V ). Further let Jopt nd Jopt e the power spent y nodes of group A nd B respectively under the miniml colition routing. Jopt = N opt (v) nd Jopt = N opt (v). v V v V Note gin tht the suscript opt to J refers to miniml colition routing for nodes of group A nd B comined. The enefit vector L corresponding to the miniml colition routing is then (L opt,l opt) where L opt = Popt Jopt nd L opt = Popt Jopt. The vector L is plotted in Fig.3 for different rndom plcements of nodes. Ech group hs nodes spred over squre of side m. If the enefit vector is in the first qudrnt (oth coordintes

A 3.5 x 3.5.5.5 under Optiml Fig. 5. Consider two sessions (,c) nd (,d). The numers next to the links re the link ndwidths. The mx-min fir ndwidth for session (,c) nd (,d) re 3 nd respectively. L c d Fig. 3..5.5.5.5.5 3 B x vectors under miniml colition.78.78 (,) (,.5). (,.5) (-,) (,) Fig.. Fir colition routing when ech node sends Mps. The numers next to the links re the rtes. re positive), then the groups mutully enefit from eing merged, otherwise one of the groups is loser. Most pirs of groups enefit from miniml colition, ut there re mny instnces in which only one group enefits. Even when pir of groups mutully enefits, there is often some disproportion in the extent of enefit, with one group getting somewht more thn the other. Theorem : The set of fesile enefit vectors is convex nd closed. E. Mx-min Fir Vector Definition 5: A fesile enefit vector B r is mx-min fir if i, B r i cnnot e incresed while mintining fesiility without decresing B j r for some group j, for which B j r Bi r. Corollry : The mx-min fir enefit vector exists nd is unique. The corollry follows s consequence of Theorem nd results from [9]. Definition 6: A Fir Colition routing is joint routing tht results in mx-min fir enefit vector. In Fig. the mx-min fir enefit vector is (.9,.9). This is chieved when node sends.78mps to nd.mps directly to AP like in Fig.. Proposition : Let r e fir colition routing. Then min(b r,b r ). Thus colition does not increse the power consumption of ny group if fir colition routing is used. Theorem : A colition will e useful if nd only if it is useful with fir colition routing r. Theorem presents necessry nd sufficient condition for deciding whether the colition would e useful. Theorem 3: For two groups the mx-min fir enefit vector hs equl components. Theorem 3 will e used in developing n efficient lgorithm for computing fir colition routing for two groups. Note tht for other resource lloction prolems e.g. ndwidth lloction, the mx-min fir vector need not hve equl components even for two contenders (Fig.5) [5]. Theorem 3 does not hold for fir power lloction lthough it holds for fir enefit lloction [9]. IV. FAIR COALITION ALGORITHM(FC) A. Description We show tht the fir colition routing nd the ssocited enefit vector cn e computed y solving the following liner progrm. FC: Mximize Z: Suject to: Z B r, Z B r, r(v, v )=O(v)+ r(v,v) v, v V V. v V V {} v V V where B r = P opt J r nd B r = P opt J r. J r = K r(v, v )d(v, v ). J r = K v V v V V {} v V v V V {} r(v, v )d(v, v ). Let Z e the ojective function vlue otined from FC. Theorem : The routing r otined s solution of FC is fir colition routing. The liner progrm involves V + constrints nd E + vriles. Hence the mx-min fir enefit vector nd the fir colition routing re polynomil complexity computle [3]. B. Simultion Results We investigte the efficcy of fir colition routing through simultions using MATLAB. Specificlly we will e interested in compring the performnce of fir colition routing with the miniml colition routing. The vlue of K depends on the choice of the wireless interfce, nd its effect is to scle our mesurements. Thus without loss of generlity we consider K =. We will lter mention detils for specific interfce. We consider squre of side m. The exit point is t the center of the squre. We consider fully connected network in which ech node cn trnsmit directly to every other node. We consider colition of two groups in rndom topologies. In Fig.6(), we investigte the cse when oth the comining groups hve equl numer of nodes. Nodes of the comining groups re uniformly distriuted over the squre re. The mx-min fir enefit vector will lwys hve equl components in this cse. We verge over the mximum component of the optiml nd the minimum of the optiml over ll the topologies. As expected the mx-min group enefit lies etween the mximum nd the ()

x 6 8 Uniform Distriution mx opt mx min x 6 8 mx opt mx min Uniform Distriution 6 x 7 5 mx opt mx min Norml Distriution 6 6 3 5 5 () Equl group sizes 5 5 3 35 5 5 () Skewed Group Sizes 5 5 3 35 5 5 (c) Clustered Groups 6 x 6 5 3 Norml Distriution mx opt mx min Totl Power 8 x 6 6 8 6 Totl Power Consumed Optiml Mx min Difference Sttisticl Mximum.8.6...8 Node Power Sttistics Optiml Mx min.6 5 5 3 35 5 5 (d) Clustered Topology 5 5 (e) Firness Overhed. 5 5 3 35 5 5 (f) Effect on Lifetime Fig. 6. Results for two groups spred over squre of side m. Mx-opt denotes the verge over the mximum component of the optiml nd min-opt denotes the verge over the minimum component of the optiml. The verge is over rndom topologies minimum components of the optiml. The enefit otined for group merger is less pronounced for sizes more thn 5-6 nodes. Therefore we will consider networks of size up to 5 nodes susequently. Fig.6() shows the results for unequl group sizes. One group is four times lrger thn the other. The smller group hs lesser enefit under the optiml in this cse. The remining trends re the sme s in the previous cse. Fig.6(c) studies the effect of clustered topologies on the enefit vlues. Nodes of ech group re normlly distriuted round rndomly chosen center. In ech cse the group with the center closer to the exit point hs negtive enefit. This group will suffer under colition routing ut in the mx-min fir cse it hs zero enefit, nd hence it does not lose. In Fig.6(d) we consider similr clustered topology where the clusters include nodes from oth groups. The trends re similr to Fig.6(c), ut oth groups otin positive enefits under fir colition. Fig.6(e) plots the totl power spent under the miniml colition routing, fir colition routing nd their difference. This difference cn e looked upon s the cost for providing firness. The verge cost is modest(8%) considering the enefit(6%) otined nd the firness chieved. Oftentimes lifetime of network is determined y the node tht spends the mximum power [3], [], [5]. Thus in Fig.6(f) we plot the quntity ( X +σ x )/ X where X is the The cost % is otined from Fig.6(e). The enefit % is with respect to the totl power consumed prior to the colition nd is otined from Fig.6() nd Fig.6(e). men power over ll nodes nd σ x is the stndrd devition. Note tht this quntity is mesure of the sttisticl mximum of the power spent y ny node. Fir colition routing hs lower vlue of this quntity s compred to the miniml. For the Lucent 8. Orinoco crd, rte of Mps in closed environment corresponds to 5dBm of output power [6]. The constnt K is then roughly 5.5 6 W/Mit m. This trnsltes to enefit of 3 Wtts for group with nodes for the uniform cse with equl group sizes. It is lso notle tht the CPU time to compute FC, for ny considered topology ws not more thn.5secs on 7Mhz/56MB RAM lptop using simplex lgorithm implementtion []. V. DISTRIBUTED IMPLEMENTATION The lgorithm in Section IV-A for computing the fir colition routing requires centrlized computtion t the exit point. Though the simplest solution, it will not e computtionlly trctle when the exit points hve cpility similr to the nodes themselves. Consider for exmple sensor network where group of sensors communicte their mesurements to common node which in turn trnsmits to sy stellite. Here we would not wnt to overwhelm the rely node with the liner progrmming computtion. Insted it would e eneficil to hve distriuted implementtion where every node performs some simple itertive computtion nd the vlues converge to the mx-min fir solution. The itertive pproch hs een motivted y

recently proposed solutions for optimiztion prolems in other resource lloction settings [], []. A. Itertive Algorithm Now we present n itertive pproch to compute fir colition routing for two groups. Let Z n nd r n denote the corresponding quntities in itertion n, where Z nd r cn e ritrrily chosen. The initil choices need not stisfy ny of the constrints. Thus ech node cn select the initil vlues of the lods for ech of its outgoing edges without ny co-ordintion with the other nodes. Similrly Z is selected t the exit point. Now we define some indictors. The enefit indictor of group is if Z n is more thn the group enefit. { if ɛ Zn + J n = r n Popt if Z n + J r n >P { opt if ɛ Zn + J r n = n Popt if Z n + J r n >Popt Node congestion c v n is the difference etween the outgoing nd the sum of the originting nd incoming trffic t node v. From (), c v n = r n (v, v ) O(v)+ r n (v,v). v V V {} v V V Node congestion indictor for node v V V is { if c v n =, s v n = if c v n >, if c v n <. Node v is considered lnced, lightly loded or hevily loded s s v n is, nd - respectively. For the exit point, s n =. We present n itertive pproch using the ove indictors. Note tht s v n cn e updted t node v using the incoming rtes in the previous itertion. Now, updte of ɛ n nd ɛ n require knowledge of the totl power eing spent y the nodes of group. This informtion cn e cquired in distriuted mnner s shown in [9]. Let {δ n } e the step-sizes tht stisfy lim n δ n = nd n= δ n =. For exmple δ n = /n stisfies the conditions. Ech node updtes its outgoing trffic s follows. [ ] + denotes the projection on [, ). r n+ (v, v )=[r n (v, v ) γδ n (s v n s v n + d(v, v ) ɛ n)] + if v V. r n+ (v, v )=[r n (v, v ) γδ n (s v n s v n + d(v, v ) ɛ n)] + if v V. The exit point updtes Z s follows. Z n+ =[Z n + δ n ( γ(ɛ n + ɛ n))] +. Theorem 5: For ll γ> the itertive procedure stted ove will converge to the mx-min fir enefit vector nd fir colition routing, irrespective of the initil choice of the itertes. VI. CONCLUSIONS We hve studied the prolem of forming colitions etween groups of nodes with the intent of sving power. We found tht n ppliction of mx-min fir techniques to this prolem yields n efficient nd lnced pproch which we cll fir colition routing. We developed theory nd lgorithms for fir colition routing. We hve crried out rnge of simultions tht demonstrte tht fir colition routing is prcticl nd eneficil in common cses. We generlize the frmework nd the computtion lgorithms for colition mong multiple groups in [9]. The colition routing lgorithms presented in this pper provide foundtions for developing opertionl protocols. Design of such protocols would require deployment of mechnisms to enforce group routings e.g., security checks. We discuss some of these issues in [9]. Acknowledgements. We pprecited comments from Ron Brchmn nd Bo Hickok. REFERENCES [] Dimitri Bertseks nd Roert Gllger. Dt Networks. Prentice Hll, 99. [] S. Cpkun, M. Hmdi, nd J. P. Huux. 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