Channel-relay price pair: towards arbitrating incentives in wireless ad hoc networks

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1 WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wrel. Commun. Mob. Comput. 2006; 6: Publshed onlne n Wley InterScence ( DOI: /wcm.383 Channel-relay prce par: towards arbtratng ncentves n wreless ad hoc networks Yuan ue 1, *,y, Baochun L 2 and Klara Nahrstedt 1 1 Department of Computer Scence, Unversty of Illnos at Urbana-Champagn, Illnos, USA 2 Department of Electrcal and Computer Engneerng, Unversty of Toronto, Toronto, Canada Summary Cooperaton n wreless ad hoc networks has twofold mplcatons. Frst, each wreless node does not excessvely and greedly nject traffc to the shared wreless channel. Second, ntermedate nodes voluntarly relay traffc for upstream nodes towards the destnaton at the cost of ts own prvate resource. Such an assumpton supports almost all exstng research when t comes to protocol desgn n ad hoc networks. We beleve that wthout approprate ncentve mechansms, the nodes are nherently selfsh (unwllng to contrbute ts prvate resource to relay traffc) and greedy (unfarly sharng the wreless channel). In ths paper, we present a prce par mechansm to arbtrate resource allocaton and to provde ncentves smultaneously such that cooperaton s promoted and the desred global optmal network operatng pont s reached by convergence wth a fully decentralzed self-optmzng algorthm. Such desred network-wde global optmum s characterzed wth the concept of Nash barganng soluton (NBS), whch not only provdes the Pareto optmal pont for the network, but s also consstent wth the farness axoms of game theory. We smulate the prce par mechansm and report encouragng results to support and valdate our theoretcal clams. Copyrght # 2006 John Wley & Sons, Ltd. KEY WORDS: prce; game theory; ad hoc network; wreless network 1. Introducton Nodes n wreless ad hoc networks not only share the wreless channel n the same local neghborhood, but also relay traffc so that destnatons multple hops away may be reached. In almost all prevous work related to wreless ad hoc networks, the followng two fundamental assumptons are made. Frst, nodes do not excessvely nject traffc to the locally shared wreless channel. Second, ntermedate nodes voluntarly relay traffc for upstream nodes towards the destnaton. In ths paper, we beleve that such assumptons may not hold n realstc scenaros, at least not wthout approprate ncentve-based mechansms. In fact, they behave n qute the contrary fashon: they are both greedy when t comes to sharng publc resource (wreless channel) and selfsh when t comes to contrbutng prvate resource (such as battery energy). In other words, the network may fal to functon at all n realstc scenaros once nether assumpton holds. The only way to solve these problems s to desgn approprate ncentve mechansms to not only encourage cooperatve behavor of selfsh nodes, but also curb unfar and excessve resource usage when sharng a common resource pool, such as a *Correspondence to: Yuan ue, Department of Electrcal Engneerng and Computer Scence, Vanderblt Unversty, Tennesse, USA. y E-mal: yuan.xue@vanderblt.edu. Copyrght # 2006 John Wley & Sons, Ltd.

2 236 Y. UE, B. LI AND K. NAHRSTEDT shared channel. Such desgns of ncentves should optmze towards a clearly specfed objectve, whch s a desred optmal operatng pont of the wreless network. At such an optmal pont, resources are shared farly, and the levels of cooperaton are adequate for all necessary data communcatons and network functons. Ths paper exactly targets ths crtcal ssue n mult-hop wreless networks. Our orgnal contrbutons are twofold. Frst, we clearly characterze the desred network-wde optmal operatng pont usng a game theoretc framework, based on the concept of Nash barganng soluton (NBS). NBS naturally encapsulates two favorable propertes: (1) Pareto effcency n terms of resource usage; and (2) a set of farness axoms wth respect to resource allocatons. Usng ths framework, the problem of fndng the desred globally optmal operatng pont may be formulated as a non-lnear optmzaton problem. Second, we propose a decentralzed algorthm that uses a prce par mechansm to arbtrate ncentves. Wth a par of prces, localzed self-optmzaton by ndvdual nodes naturally converges to globally optmal network operatng ponts. Wthn the prce par, the channel prce regulates greedy usage of the shared wreless channel, whle the relay prce encourages traffc relayng. Effectvely, our prce par mechansm transforms non-cooperatve behavor n wreless ad hoc networks to a cooperatve game, whose optmal operatng ponts demonstrate more advantageous propertes than the usual Nash equlbrum n typcal non-cooperatve envronments. The essence of our paper s to ntegrate the mechansms that use prcng as sgnals to (1) farly allocate resources; and (2) adequately ncentvze cooperatve behavor. Though there exsts prevous work towards ether one of these objectves, we are not aware of exstng work that ntegrates both prces nto a coherent framework. Such ntegraton becomes more complcated f we consder the unque channel contenton characterstcs n wreless ad hoc networks, where the traffc flows contend n multple contenton clques. Consderatons of such unque complcatons n ad hoc networks are beyond all of the exstng work n the area of prcng or ncentves. The remander of ths paper s organzed as follows. Secton 2 presents some prelmnares before formal treatment of ths topc. Secton 3 defnes the desred network operatng ponts usng the concept of NBS. We present the dstrbuted algorthm n Secton 4. We show smulaton results n Secton 5, present related work n Secton 6, and fnally conclude the paper n Secton Network Model We consder a wreless ad hoc network whch conssts of a set of nodes N ¼ f1; 2;... ; Ng. In ths network, only nodes that are wthn the transmsson range of each other can communcate drectly and form a wreless lnk. We model such a network as a bdrectonal graph G N ¼ ðn ; LÞ, where L ¼ f1; 2;... ; Lg s the set of wreless lnks. In such a network, a wreless node 2 N may establsh an end-to-end flow, or smply flow, f wth rate x to another node. Flow f s assumed to be elastc: t requres a mnmum rate of x m and a maxmum rate of x M, that s, x m 4x 4x M. In general, f flows through multple hops n the network, passng a set of wreless lnks. We use ths set of wreless lnks to represent f, that s, f L. We denote the set of relayng nodes for flow f as Rð f Þ, and the destnaton of f as Dð f Þ. For smplcty of exposton, we further defne Hð f Þ ¼ Rð f Þ [ fdð f Þ; g as the set of nodes f traverses, and Kð f Þ ¼ Hð f Þ fg. A sngle-hop data transmsson along a partcular wreless lnk s referred to as a subflow and s a part of a flow. Several subflows from dfferent flows along the same wreless lnk form an aggregated subflow. In such a network, nodes compete for two types of resources: shared wreless channel and ndvdual nodes relayng cost (such as energy). The avalablty of these resources constrans the soluton space of resource allocatons. We proceed to analyze the characterstcs of both types of resources Shared Wreless Channel: Locaton-Dependent Contenton The shared-medum mult-hop nature of wreless ad hoc networks presents unque characterstcs of locatondependent contenton and spatal reuse of spectrum. Compared wth wrelne networks where flows contend only at the router wth other smultaneous flows through the same router (contenton n the tme doman), the unque characterstcs of mult-hop wreless networks show that flows also compete for shared channel bandwdth f they are wthn the transmsson ranges of each other (contenton n the spatal doman). In partcular, two subflows contend wth each other f ether the source or destnaton of one subflow s wthn the transmsson range of the source or Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

3 CHANNEL-RELAY PRICE PAIR 237 destnaton of the other. y The localty of wreless transmssons mples that the degree of contenton for the shared medum s locaton dependent. On the other hand, two subflows that are geographcally far away have the potental to transmt smultaneously, reusng the wreless channel. We now formulate the resource constrants that reflect the unque characterstcs of wreless ad hoc networks. Frst, let us consder a set of mutually contendng subflows. In ths set, only one subflow can transmt at a tme. Intutvely, the aggregated rate of all subflows n ths set can not exceed the channel capacty. Formally, we consder a subflow contenton graph. In ths graph, each vertex corresponds to an aggregated subflow n the orgnal network. Each edge n the graph denotes that two aggregated subflows whch correspond to the two vertces, contend wth each other. Formally, let V C ¼ [ 2N f L be the set of aggregated subflows n network G N, then a bdrectonal graph G C ¼ ðv C ; E C Þ s a subflow contenton graph of network G N. In a graph, a complete subgraph s referred to as a clque. A maxmal clque s defned as a clque that s not contaned n any other clques. z In a subflow contenton graph, the vertces n a maxmal clque represent a maxmal set of mutually contendng subflows. Intutvely, each maxmal clque n a subflow contenton graph represents a maxmal dstnct contenton regon, snce at most one subflow n the clque can transmt at any tme and addng any other subflows nto ths clque wll ntroduce the possblty of smultaneous transmssons. For smplcty, we use the set of vertces n a clque to represent the clque, and denote t as q. Furthermore, we denote the set of all maxmal clques n a subflow contenton graph as Q. Here, we llustrate the above concepts usng an example. The network topology and the flows n the example are shown n Fgure 1(a). The correspondng subflow contenton graph s shown n Fgure 1(b). In ths example, there are three maxmal clques n the contenton graph: q 1 ¼ ff1; 2g; f3; 2g; f3; 4g; f3; 6gg; q 2 ¼ ff3; 2g; f3; 4g; f4; 5g; f3; 6gg and q 3 ¼ ff3; 2g; f3; 4g; f3; 6g; f6; 7gg. We proceed to consder the problem of allocatng rates to wreless lnks. We clam that a rate allocaton y If we assume that the nterference range s greater than the transmsson range, the contenton model can be straghtforwardly extended. z Note that maxmal clque has a dfferent defnton from maxmum clque of a graph, whch s the maxmal clque wth the largest number of vertces. Fg. 1. Example of wreless ad hoc network and ts subflow contenton graph. (a) Network topology and (b) Subflow contenton. y ¼ ðy l ; l 2 LÞ s feasble, f there exsts a collsonfree transmsson schedule that allocates y l to l. We now formalze the condton mpled by such a feasble rate allocaton. Lemma 1. If a rate allocaton y ¼ ðy l ; l 2 LÞ s feasble, then the followng condton s satsfed: 8q 2 Q; y l 4C l2q ð1þ where C s the channel capacty. Equaton (1) gves an upper bound on the rate allocatons to the wreless lnks. In practce, however, such a bound may not be tght, especally wth carrer-sensng-multple-access-based wreless networks (such as IEEE ). In ths case, we ntroduce C q, the achevable channel capacty at a clque q. More formally, f P l2q y l4c q then y ¼ ðy l ; l 2 LÞ s feasble. To ths end, we observe that each maxmal clque may be regarded as an ndependent channel resource unt wth capacty C q. It motvates the use of the maxmal clque as a basc resource unt for prcng n wreless ad hoc networks, as compared to the noton of a lnk n wrelne networks. We now proceed to consder resource constrants on rate allocatons among flows. To facltate dscussons, we defne a clque-flow matrx R ¼ fr q g, where R q ¼ jq \ f j represents the number of subflows that flow f has n the clque q. If we treat a maxmal clque as an ndependent resource, then the clque-flow matrx R represents the resource usage pattern of each flow. Let the vector C ¼ ðc q ; q 2 QÞ be the vector of achevable channel capactes n each of the clques. Constrants wth respect to rate allocatons to end-to-end flows n wreless ad hoc networks are presented n the followng proposton. Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

4 238 Y. UE, B. LI AND K. NAHRSTEDT Proposton 1. In a wreless ad hoc network G N ¼ ðn ; LÞ, there exsts a feasble rate allocaton x ¼ ðx ; 2 N Þ, f and only f R x4c. P Proof: It s obvous that R x4c, 8q 2 Q, P2N R q x 4C q. By the defnton of R, we have 2N R q x ¼ P l2q y l. The result follows naturally from Lemma 1 and ts followng dscussons. & 2.2. Costs of Relays Relayng traffc for upstream nodes apparently ncurs cost, snce localzed resources need to be consumed, such as energy, CPU cycle, and memory space. Wthout loss of generalty, we use energy levels on each node as an example to characterze such costs of relays. Gven a mnmum expected lfetme n the network, each node j has a budget on ts energy consumpton rate, denoted as E j. Here, we consder two types of energy consumpton related to packet transmsson: (1) e r as the energy consumed for recevng a unt flow; (2) e s as the energy consumed for transmttng a unt flow. Then the energy consumpton at node j s x j e s þ P :j2rðf Þ x ðe r þ e s Þ þ P :j¼dð f Þ x e r. As the energy consumpton rate can not exceed the energy budget, we have the followng relaton: x j e s þ :j2rð f Þ x ðe r þ e s Þ þ :j¼dð f Þ x e r 4E ð2þ We now proceed to defne a N N matrx B as follows. 8 cre s f j ¼ e r þ e s f node j forwards packets for flow f ; >< that s; j 2 Rð f Þ B j ¼ e r f nodej s the destnaton of flow f ; that s; j ¼ Dð f Þ >: 0 otherwse B specfes the relayng relaton among nodes n the ad hoc network. To summarze, the local constrant on energy can be formalzed as follows: B x E ð4þ where E ¼ ðe j ; j 2 N Þ s the energy consumpton budget vector. 3. Problem Formulaton In ths secton, we characterze the desred networkwde optmal operatng pont usng a game theoretc framework, based on the concept of NBS. NBS naturally encapsulates two favorable propertes: (1) Pareto effcency n terms of resource usage; and (2) a set of farness axoms wth respect to resource allocatons. Usng ths framework, the problem of fndng the desred globally optmal operatng pont may be formulated as a non-lnear optmzaton problem. We show how such a global optmzaton problem may be decomposed nto localzed greedy optmzaton problems va a prce par Nash Barganng Soluton: a Game Theoretcal Formulaton We present the basc concepts and results of NBS from game theory [1] and show how t can characterze and formulate the desred network operaton pont, towards whch our prce par mechansm should converge. The basc settng of the problem s as follows: the set of nodes N n the wreless ad hoc network G N consttutes a set of players n the game. They compete for the use of a fxed amount of resources (wreless channel and costs of relays such as energy). The rate allocaton x ¼ ðx ; 2 N Þ s the utlty vector of all players n the game. Let S R N be the set of all feasble utlty vectors. We assume that S s a non-empty convex closed and bounded set. Further, we denote the ntal agreement pont of the game as x, whch s the guaranteed utltes of players wthout any cooperaton n order to enter the game. In such a problem settng, a barganng problem s any ðs; x Þ where fx 2 Sjx x g 6¼ ;. In other words, a barganng problem s actually a set of rate allocatons x that s acceptable to all players. Let B ¼ fðs; x Þg denote the set of all barganng problems. It then follows that a barganng soluton s any functon : B! R N, so that 8ðS; x Þ 2 B, ðs; x Þ 2 S. A barganng soluton actually specfes fndng a rate allocaton wthn all acceptable allocatons. A natural queston about the barganng soluton s how such a functon s to be defned. There are two reasonable propertes desred for a barganng soluton: (1) effcent use of resources; and (2) far allocaton among all players. These two condtons are precsely encapsulated by the concept of NBS defned as follows. Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

5 CHANNEL-RELAY PRICE PAIR 239 Defnton 1 (Nash Barganng Soluton). A barganng soluton : B! R N s a NBS, f x ¼ ðs; x Þ satsfes the Nash Axoms A1 A6. A1 (ndvdual ratonalty) x x ; A2 (feasblty) x 2 S; A3 (pareto optmalty), If 8x 2 S; x x, then x ¼ x; A4 (ndependence of rrelevant alternatves) If x 2 T S and x ¼ ðs; x Þ, then x ¼ ðt; x Þ; A5 (ndependence of lnear transformaton) Let T be obtaned from S by the lnear transformaton ðxþ wth ðxþ ¼ a x þ b ; a ; b > 0; ¼ 1; 2;... ; N Then f ðs; x Þ ¼ x, then ðt; ðx ÞÞ ¼ ðxþ; A6 (symmetry) Suppose S s symmetrc wth respect to a subset J f1; 2;... ; Ng of ndces, then s symmetry, whch means that f x 2 S and for ; j 2 J, x ¼ x j, then x x ¼ xx j. The above axoms encapsulate both the concept of Pareto optmalty (A3) and the concept of farness (A4 A6). Pareto optmalty means that there s no other pont whch gves strct superor utlty for all the players smultaneously. The defnton of Pareto optmalty reflects the condton of effcent use of resources, where there are no dle resources n the network. The exstng work n game theory [1] and ts applcaton n wrelne communcaton networks [2] establsh the followng results for NBS. Proposton 2. There exsts a unque functon defned on all barganng problems ðs; x Þ that satsfes axoms A1 A6, that s, a unque NBS exsts. Moreover, the unque soluton x s a unque vector that solves the followng maxmzaton problems: Equvalently, N 1 : max x N 2 : max x Y ðx x Þ 2N lnðx x Þ 2N ð5þ ð6þ the mnmum rate vector x m can be regarded as an ntal agreement pont of the game x. From the dscussons n Secton 3A, t s clear that the NBS formulates the desred network operaton pont, whch s far to all nodes whle effcent from the network s pont of vew. Thus, the problem of fndng the globally optmal resource allocaton s transformed to solve the NBS of ts correspondng game, whch s the soluton of the followng non-lnear optmzaton problem by Proposton 2. P : maxmze lnðx x m Þ 2N ð7þ subject to A x C ð8þ B x E ð9þ over x m x x M ð10þ The objectve functon n Equaton (7) of the optmzaton problem corresponds to the optmzaton problem whose soluton s NBS. The constrants of the optmzaton problem Equatons (8) and (9) are the constrants from the shared wreless channel and costs of relays, respectvely, as dscussed n Secton 2. Note that the objectve functon P 2N lnðx x m Þ s strctly concave. In addton, the feasble regon of the optmzaton problem s non-empty, convex, and compact. By non-lnear optmzaton theory, there exsts a maxmzng value of argument x for the above optmzaton problem. Let us consder the Lagrangan form of the optmzaton problem P: Lðx; l ; l Þ ¼ 2N lnðx x m Þ þ l ðc AxÞ þ l ðe BxÞ ð11þ where l ¼ ð q ; q 2 QÞ, l ¼ ð j ; j 2 N Þ are two vectors of Lagrange multplers. The frst-order Kuhn Tucker condtons are necessary and suffcent for optmalty of problem P. Thus, for 2 N, the followng condtons hold: 1 x x m q A q j B j ¼ 0 q2q j2n ð12þ 3.2. Network Operatng Ponts: Optmal and Far Rate Allocatons We assume that each player nvolved n the game can be guaranteed wth ts mnmum rate vector x m. Thus, l ðc AxÞ ¼ 0; l 0 l ðe BxÞ ¼ 0; l 0 x m x x M ð13þ ð14þ ð15þ Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

6 240 Y. UE, B. LI AND K. NAHRSTEDT In the Lagrangan form shown n Equaton (11), the Lagrange multplers q can be regarded as the mpled cost of unt flow accessng the channel at the maxmal clque q. In other words, q s the shadow prce of clque q, called channel prce. Ths prce corresponds to the shared channel resource constrant. The Lagrange multplers j can be regarded as the mpled relay cost of unt flow at node j. In other words, j s the shadow prce of relay at node j, called relay prce. Ths par of prces ðl ; l Þ wll be used as ncentves so that localzed self-optmzng decson can mplement the global optmum. In partcular, the prce l wll sgnal a charge (contrary to ncentves) to the shared channel usage and regulate the greedy behavor, whle the prce l wll provde ncentves to traffc relays at ntermedate nodes and regulate the selfsh behavor of wreless nodes. Let us denote Clearly, ¼ q2q q A q ð16þ ¼ j2n ¼ ¼ es þ j2rð f Þ q:f \q6¼; j B j ¼ l:l2f ¼ j2hð f Þ q A q q q:l2q j B j j ðes þ e r Þ þ Dð f Þ er ð17þ ð18þ ð19þ ð20þ ð21þ Then and are the prces for node, whch s the source of flow f, for accessng shared channels and relay servces, respectvely. For channel usage, node needs to pay for all the maxmal clques that t traverses. For each clque, the prce to pay s the product of the number of wreless lnks that f traverses n ths clque and the shadow prce of ths clque as n Equaton (18). Alternatvely, the prce of Fg. 2. An example of the prce par mechansm. (a) Channel prce and (b) relay prce. flow f s the aggregated prce of all ts subflows. For each subflow, ts prce s the aggregated prce of the maxmal clques that t belongs to as n Equaton (19). Note that such a prcng polcy for end-to-end flows s fundamentally dfferent from the prcng models n wrelne networks, where a flow s prce s the aggregaton of the lnk prces whch t traverses. Such dfference s rooted at the nature of the shared medum the nterference and spatal reuse of wreless channel n an ad hoc network. For traffc relay servces, node needs to pay for the relay costs of all relayng nodes of f, ncludng tself and the destnaton. Usng flow f 1 as an example, the channel prce model s llustrated n Fgure 2(a) and the relay prce model s llustrated n Fgure 2(b). The channel prce for f 1 s 1 ¼ 3 1 þ 3 2 þ 2 3 ; and the relay prce 1 ¼ 2 1 þ 3 2 þ 3 3 þ 3 4 þ 5, where the energy consumed for recevng s gven as e r ¼ 1, and for transmttng as e s ¼ 2. To summarze, we have the followng results wth respect to the globally optmal network operatng pont. Theorem 1. There exsts a unque soluton to problem P (.e., unque NBS), whch s characterzed as follows: There exst two vectors l ¼ ð q ; q 2 QÞ, l ¼ ð j ; j 2 N Þ such that " # x M 1 x ¼ þ þ x m ; for 2 N ð22þ l ðc AxÞ ¼ 0; l 0 l ðe BxÞ ¼ 0; l 0 where ½zŠ b a ¼ maxfmnfz; bg; ag. x m ð23þ ð24þ Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

7 CHANNEL-RELAY PRICE PAIR Local Strateges: Self-Optmzng Decsons Wth the understandng of the global optmal pont, we now study how ths pont can be acheved n a dstrbuted manner by localzed self-optmzng decsons at each ndvdual node. The key to ths goal s to use the par of prces ðl ; l Þ as sgnals to coordnate the dstrbuted decsons. Frst, we study the condtons that the prces need to satsfy n order to ncentvze local node decsons so that they could mplement the global network optmum. Let us consder the followng problems: ChannelðA; C; l Þ: maxmze x 2N ð25þ subject to Ax C ð26þ over x m x x M ð27þ where s a functon of l, as defned n Equaton (16). Problem ChannelðA; C; l Þ maxmzes the total revenue of channel based on chargng per unt of bandwdth to user subject to the channel capacty constrant. RelayðB; E; l Þ: maxmze x ð28þ 2N subject to Bx E ð29þ over x m x x M ð30þ where s a functon of l, as defned n Equaton (17). Problem RelayðB; E; l Þ maxmzes the total relay revenue of all nodes subject to the relay cost constrant at each ndvdual node. Now let us consder a local self-optmzng decson at each node whch corresponds to the followng problem: Node ð ; ; Þ: maxmze lnðx x m Þ x x þ E ð31þ over x m x x M ð32þ The relatonshp between localzed node decson and the global optmal network operatng pont s then gven as follows. Theorem 2: There exst vectors l, l and x such that (1) x s the unque soluton to Node ð ; ; Þ; (2) x solves ChannelðA; C; l Þ; (3) x solves RelayðB; E; l Þ; Then x also solves problem P. The proof of ths theorem s gven n our techncal report [3]. Let us denote ðx Þ ¼ lnðx x m Þ x x þ E ð33þ Now we show that ðx Þ reflects the net beneft of node and problem Node ð ; ; Þ maxmzes the node s net beneft. Ths clam s based on the followng two observatons. Frst, node does not pay ts relay cost to tself from a local pont of vew, whle the flow relay prce, whch s defned from a global pont of vew, contans the prce of relay for tself. Thus from a local pont of vew, the cost of node s: x þ B j ¼ ð þ Þx es x ð34þ j2kð f Þ And the revenue of node s: j:2kð f j Þ B jx j ¼ j:2hð f j Þ B j x j es x Second, from Theorem 2 we have ¼ 0; f B j x j < E > 0; f j:2hð f j Þ j:2hð f j Þ B j x j ¼ E ð35þ ð36þ ð37þ Thus t s clear that ðx Þ reflects the net beneft of node, whch s the dfference between utlty, revenue 3 and cost. 4. Algorthm Although the global problem P can be mathematcally solvable n a centralzed fashon, t s mpractcal for realstc operatons n wreless ad hoc networks. In Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

8 242 Y. UE, B. LI AND K. NAHRSTEDT ths secton, we present a dstrbuted teratve algorthm for prce calculaton and resource arbtraton. We show that the teratve algorthm converges to the global optmum, and maxmzes the local net beneft at each node smultaneously Dual Problem In order to acheve a dstrbuted soluton, we frst look at the dual problem of P as follows. where D : mn l ;l 0 Dðl ; l Þ ð38þ adjusted n the opposte drecton to the gradent rdðl ; l Þ: q ðt þ 1Þ ¼ ½ q ðtþ; l Š þ q j ðt þ 1Þ ¼ ½ j ðtþ; l Š þ j ð41þ ð42þ where s the stepsze. Dðl ; l Þ s contnuously dfferentable snce lnðþ s strctly concave [4]. Thus, t follows that Dðl ; l Þ ¼ max x ¼ max x Lðx; l ; l Þ ðlnðx x m Þ ð þ Þx þ E Þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2N ðx Þ þ q2q q C ; l q ¼ C ; l Þ ¼ E j :f \ q6¼; :j2hð f Þ x ð ; ÞA q x ð ; ÞB j ð43þ ð44þ Note that ðx Þ s node s net beneft. By the separaton nature of Lagrangan form, maxmzng Lðx; l ; l Þ can be decomposed nto separately maxmzng ðx Þ for each node 2 N (Secton n Reference [4]). We now have Dðl ; l Þ ¼ 2N max fðx Þg þ x m x x M q c q ð39þ q2q As ðþ s strctly concave and twce contnuously dfferentable, a unque maxmzer of ðx Þ exsts when dðx Þ ¼ 1 dx x x m ð þ Þ ¼ 0 We defne the maxmzer as follows: x ð ; Þ ¼ arg max fðx Þg x m x x M ð40þ Note that x ð ; Þ s usually called demand functon, whch reflects the optmal rate for node wth channel prce as and relay prce as Dstrbuted Algorthm We solve the dual problem D usng the gradent projecton method [4]. In ths method, l and l are Substtutng Equaton (43) nto (41) and (44) nto (42), we have q ðt þ 1Þ ¼ ½ q ðtþ þ ð x ð ðtþ; ðtþþa q C q ÞŠ þ :f \q6¼; j ðt þ 1Þ ¼ ½ j ðtþ þ ð :j2hð f Þ ð45þ x ð ðtþ; ðtþþb j E j ÞŠ þ ð46þ Equatons (45) and (46) reflect the law of supply and demand. If the demand for bandwdth at clque q exceeds ts supply C q, the channel constrant s volated. Thus, the channel prce q s rased. Otherwse, q s reduced. Smlarly, n Equaton (46), f the demand for energy at node j exceeds ts budget E j, the energy constrant s volated. Thus, the relay prce j s rased. Otherwse, j s reduced. We summarze our algorthm n Table I, where clque q and node are deemed as enttes capable of computng and communcatng. We now show the property of ths dstrbuted teratve algorthm. Let us defne YðÞ ¼ P q A q þ P j B j, and Y ¼ max 2N YðÞ. Further, we defne UðqÞ ¼ P 2N A q Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

9 CHANNEL-RELAY PRICE PAIR 243 Table I. Dstrbuted algorthm. Clque prce update (by clque q): at tme t ¼ 1,2,... 1 Receve rates x ðtþ from flows f where f \ q 6¼ ; 2 Update prce q ðt þ 1Þ ¼ ½ q ðtþ þ ðp f\q6¼; x ðtþa q C q ÞŠ þ 3 Send q ðt þ 1Þ to flows f where f \ q 6¼ ; Relay prce update (by node j): at tme t ¼ 1,2,... 1 Receve rates x ðtþ from flows f where j 2 Hðf Þ 2 Update prce j ðt þ 1Þ ¼ ½ j ðtþþ ðp :j2hðfþ x ðtþb j E j ÞŠ þ 3 Send j ðt þ 1Þ to flows f where j 2 Hð f Þ Rate update (by node ): at tme t ¼ 1; 2;... 1 Receve prces q ðtþ from q where f \ q 6¼ ; 2 Receve relay prces j ðtþ from j where j 2 Hðf Þ 3 Calculate ¼ P q:f\q6¼; q A q ¼ P j:j2hðfþ l j B j 5 Adjust rate x ðt þ 1Þ ¼ x ð ; Þ 4 Send x ðt þ 1Þ to correspondng clques. and U ¼ max q2q UðqÞ; Vð jþ ¼ P 2N B j and V ¼ max j2n Vð jþ; Z ¼ maxf U; Vg. Let ¼ ðx M x m Þ 2 and ¼ max 2N. Theorem 3 (Global convergence and optmalty). Suppose 0 < < 2=Y Z. Startng from any ntal rates x m xð0þ x M, and prces l ð0þ 0 and l ð0þ 0, every accumulaton pont ðx ; l ; l Þ of the sequence ðxðtþ; l ðtþ; l ðtþþ generated by the algorthm n Table I s prmal-dual optmal. The reader s referred to our techncal report [3] for a detaled proof. Though there exsts a unque maxmzer x to the problem P, there may be multple dual optmal prces, snce only the flow prce s constraned at optmalty accordng to Uf 0ðx f Þ ¼ þ. Theorem 3 does not guarantee convergence to a unque vector ðx ; l ; l Þ, though any convergent subsequence leads to the optmal rate allocaton x. In the above teratve algorthm, a maxmal clque s regarded as a network element that can carry out the functons of prce calculaton and notfcaton. However, a maxmal clque s only a concept defned based on subflow contenton graph. To deploy the algorthm n an actual ad hoc network, the above tasks of a maxmal clque need to be carred out by the nodes that consttute the clque n a dstrbuted fashon. For the mplementaton detals, readers are referred to our report [3]. 5. Smulaton Results We present the smulaton results of our prce par mechansm and the dstrbuted algorthm on a smple network as shown n Fgure 1. The reader s referred to our techncal report [3] for extensve results and performance evaluaton. In the frst experment, the network parameters are as follows: channel capacty C q ¼ 2 Mbps, for q ¼ 1,2,3; relay cost E j ¼ 2 for j ¼ 1,2,3,5,6,7 and E 4 ¼ 3. The mnmum and maxmum rate requrement of flows are fx m ¼ 0 Mbps and x M ¼ 2 Mbps, for all ¼ 1;... ; 7. Step sze ¼ 0:05. It s obvous that the mnmum rate requrement can be guaranteed. We show the convergence behavor of our teratve algorthm n ths experment. As shown n Fgure 3, the algorthm converges to a global network equlbrum wthn about 800 teratons. At the equlbrum pont, the optmal resource allocaton and prces are lsted n Table II. In the second set of experments, we show how both channel prce and relay prce are necessary to regulate and ncentvze the network to operate at ts optmal pont. In partcular, we smulate on the same network as n the frst experment wth only one prce. The results n comparson wth the result based on prce Fg. 3. Convergence of the algorthm on the network shown n Fgure 1. (a) Flow rates x ð 2 N Þ, (b) Channel prce q ðq 2 QÞ and (c) relay prce j ð j 2 N Þ: Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

10 244 Y. UE, B. LI AND K. NAHRSTEDT Table II. Equlbrum rate comparson. Rate(Mbps) x 1 x 2 x 3 x 4 x 5 x 6 x 7 P 7 ¼1 lnðx Þ Prce par Channel prce only Relay prce only par are shown n Table II. It s easy to see that the network does not operate at ts desred pont (.e., NBS), when only one prce (ether channel prce or relay prce) s used. Above results show that n a typcal network envronment, usng ether only channel prce to regulate the usage of shared wreless channel or only relay prce to ncentvze traffc relay for other nodes can not reach the adequate level of cooperaton, where global network operates at ts optmal pont. 6. Related Work The problem of optmal and far resource allocaton has been extensvely studed n the context of wrelne networks. Among these works, prcng has been shown to be an effectve approach to acheve dstrbuted soluton for flow control [5 7] and servce dfferentaton [8]. Smultaneously, game theory s appled to model resource sharng among multple users, e.g., [2,9]. The man dfference that dstngushes our work from the exstng works s rooted n the unque characterstcs of resource models of ad hoc network. Frst, the allocaton mechansm needs to consder two types of resources, namely, the shared channel and the prvate resource such as energy. Second, due to the locaton-dependent contenton and spatal reuse of the shared channel resource, the channel prce s assocated wth a maxmal clque n the subflow contenton graph, rather than a wrelne lnk. Ths presents a dfferent prcng polcy for end-to-end flows. Incentves n wreless networks have stmulated much research nterests. (e.g., n the context of ad hoc networks [10 12] and n wreless LAN [13]). In partcular, the work n Reference [10] presents vrtual credt-based mechansms to stmulate cooperaton n ad hoc networks, where vrtual credts (so called nuglets) are awarded for packet forwardng. Some approaches [14,15] use a reputaton-based mechansm where selfsh or msbehavng nodes are dentfed, solated, or punshed. Our work dstngushes from the exstng works n that, t does not only promote cooperaton n packet forwardng, more mportantly, t studes at what level of cooperaton the network operates at ts optmal pont, and how to acheve such cooperaton usng prcng as ncentves. Non-cooperatve game theory has been used to model the relayng behavor among nodes n ad hoc networks n References [11,12]. By desgnng approprate game strateges and analyzng the Nash equlbrum of the correspondng relayng game, these works show the exstence of a network operatng pont where node cooperaton s promoted. In our work, Nash barganng soluton s used to characterze the global network operatng pont, whch usually demonstrates more advantageous propertes, such as Pareto optmalty and farness, than the usual Nash equlbrum n a non-cooperatve game. There are also prevous works that address the ssue of resource allocaton [16] and use a prce-based approach [17]. However, the ad hoc network models n these works do not consder the shared nature of the wreless channel, and thus ther solutons are not able to capture the unque ssues n wreless ad hoc networks. Moreover, the prce-based dstrbuted algorthm presented n [16] only converges to a network optmum when ts utlty functon takes certan a specal form, and such a utlty functon does not satsfy the farness axom. 7. Concludng Remarks Ths paper presents a prce par mechansm that both regulates greedy behavors and ncentvzes selfsh users n ad hoc networks. A par of prces s the centerpece of ths mechansm: (1) the channel prce that reflects the unque characterstcs of locatondependent contenton n ad hoc networks, and regulates the usage of shared wreless channel; (2) the relay prce that gves ncentves to reach the adequate Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6:

11 CHANNEL-RELAY PRICE PAIR 245 level of cooperaton wth respect to traffc relay. By usng such a prce par as a sgnal, the decentralzed self-optmzng decsons at each ndvdual node converges to the global network optmal operaton pont. References 1. Owen G. Game Theory. Academc Press: New York, USA, Yache H, Mazumdar RR, Rosenberg C. A game theoretc framework for bandwdth allocaton and prcng n broadband networks. IEEE/ACM Transactons on Networkng 2000; 8(5): ue Y, L B, Nahrstedt K. Channel-relay prce par: towards arbtratng ncentves n wreless ad hoc networks. Tech. Rep. UIUCDCS-R , UILU-ENG , Unversty of Illnos at Urbana Champagn, Bertsekas D, Tstskls J. Parallel and Dstrbuted Computaton: Numercal Methods. Athena Scentfc Belmont MA, Prentce-Hall, Kelly FP, Maulloo AK, Tan DKH. Rate control n communcaton networks: shadow prces, proportonal farness and stablty. Journal of the Operatonal Research Socety 1998; 49: Low SH, Lapsley DE. Optmzaton flow control: basc algorthm and convergence. IEEE/ACM Transactons on Networkng 1999; 7(6): Kunnyur S, Srkant R. End-to-end congeston control: utlty functons, random losses and ecn marks. In Proceedngs of INFOCOM, Cocch R, Shenker S, Estrn D, Zhang L. Prcng n computer networks: motvaton, formulaton, and example. IEEE/ACM Transactons on Networkng 1993; 1(6): Cao R, Shen H, Mlto R, Wrth P. Internet prcng wth a game theoretcal approach: concepts and examples. IEEE/ACM Transactons on Networkng 2002; 10(2): Buttyan L, Hubaux JP. Stmulatng cooperaton n selforganzng moble ad hoc networks. ACM/Kluwer Moble Networks and Applcatons 2003; 8(5): Srnvasan V, Nuggehall P, Chassern CF, Rao RR. Cooperaton n Wreless Ad Hoc Networks. In Proceedngs of IEEE INFOCOM, Urp A, Bonuccell M, Gordano S. Modelng cooperaton n moble ad hoc networks: a formal descrpton of selfshness. In Proceedngs of Workshop modelng and optmzaton n moble, ad hoc and wreless networks (WOpt), Lao R, Wouhayb R, Campbell A. Incentve engneerng n wreless lan based access networks. In Proceedngs of 10th IEEE Internatonal Conference on Network Protocols (ICNP), November, Buchegger S, Le Boudec JY. Performance analyss of the confdant protocol (cooperaton of nodes farness n dynamc ad-hoc networks. In Proceedngs of MobHoc, June Zhong S, Yang Y, Chen J. Sprte: a smple, cheat-proof, credtbased system for moble ad hoc networks. In Proceedngs of IEEE INFOCOM, Qu Y, Marbach P. Bandwth allocaton n ad-hoc networks: a prce-based approach. In Proceedngs of INFOCOM, Crowcroft J, Gbbens R, Kelly F, Ostrng S. Modellng ncentves for collaboraton n moble ad hoc networks. In Proceedngs of workshop modelng and optmzaton n moble, ad hoc and wreless networks (WOpt), Authors Bographes Yuan ue receved her B.S. n Computer Scence from Harbn Insttute of Technology, Chna n 1998 and her M.S. and Ph.D. n Computer Scence from the Unversty of Illnos at Urbana-Champagn n 2002, and Currently she s an assstant professor at the Department of Electrcal Engneerng and Computer Scence of Vanderblt Unversty. She s a recpent of the Vodafone fellowshp. Her research nterests nclude wreless and sensor networks, QoS support, and network securty. She s a member of the IEEE. Baochun L receved hs B.Engr. degree n 1995 from Department of Computer Scence and Technology, Tsnghua Unversty, Chna, and hs M.S. and Ph.D. degrees n 1997 and 2000 from the Department of Computer Scence, Unversty of Illnos at Urbana-Champagn. Snce 2000, he has been wth the Department of Electrcal and Computer Engneerng at the Unversty of Toronto, where he s currently an Assocate Professor. He holds the Nortel Networks Junor Char n Network Archtecture and Servces snce October 2003, and the Bell Unversty Laboratores Endowed Char n Computer Engneerng snce August In 2000, he was the recpent of the IEEE Communcatons Socety Leonard G. Abraham Award n the Feld of Communcatons Systems. Hs research nterests nclude applcaton-level Qualty of Servce provsonng, wreless and overlay networks. He s a senor member of IEEE, and a member of ACM. Hs emal address s bl@eecg.toronto.edu. Klara Nahrstedt Klara Nahrstedt (M 94) receved her A.B., M.Sc. degrees n Mathematcs from the Humboldt Unversty, Berln, Germany, and her Ph.D. n Computer Scence from the Unversty of Pennsylvana. She s an full professor at the Unversty of Illnos at Urbana-Champagn, Computer Scence Department where she does research on Qualty of Servce (QoS)-aware systems wth emphass on end-to-end resource management, routng, and mddleware ssues for dstrbuted multmeda systems. She s the coauthor of the wdely used multmeda book Multmeda:Computng, Communcatons and Applcatons publshed by Prentce Hall, and the recpent of the Early NSF Career Award, the Junor erox Award, and the IEEE Communcaton Socety Leonard Abraham Award for Research Achevements, and the Ralph and Catherne Fsher Professorshp Char. Snce June 2001, she serves as the edtor-n-chef of the ACM/Sprnger Multmeda System Journal. Copyrght # 2006 John Wley & Sons, Ltd. Wrel. Commun. Mob. Comput. 2006; 6: