Multipath Wireless Network Coding: A Population Game Perspective

Transcription

1 Multath Wreless Network Codng: A Poulaton Game Persectve Vnth Reddy, rnvas hakkotta, Alex rntson and Natarajan Gautam Det. of ECE, Texas A&M Unversty Det. of IE, Texas A&M Unversty Emal: {vnth reddy, sshakkot, salex, gautam}@tamu.edu Abstract We consder wreless networks n whch multle aths are avalable between each source and destnaton. We allow each source to slt traffc among all of ts avalable aths, and ask the queston: how do we attan the lowest ossble number of transmssons er unt tme to suort a gven traffc matrx? Traffc bound n ooste drectons over two wreless hos can utlze the reverse caroolng advantage of network codng n order to decrease the number of transmssons used. We call such coded hos as hyer-lnks. Wth the reverse caroolng technque longer aths mght be cheaer than shorter ones. However, there s a rsoners dlemma tye stuaton among sources the network codng advantage s realzed only f there s traffc n both drectons of a shared ath. We develo a twolevel dstrbuted control scheme that decoules user choces from each other by declarng a hyer-lnk caacty, allowng sources to slt ther traffc selfshly n a dstrbuted fashon, and then changng the hyer-lnk caacty based on user actons. We show that such a controller s stable, and verfy our analytcal nsghts by smulaton. I. INTRODUCTION There has recently been sgnfcant nterest n multho wreless networks, both as a means for basc Internet access, as well as for buldng secalzed sensor networks. However, lmted wreless sectrum together wth nterference and fadng ose sgnfcant challenges for network desgners. The technque of network codng has the otental to mrove the throughut and relablty of multho wreless networks by takng advantage of the broadcast nature of wreless medum. For examle, consder a wreless network codng scheme dected n Fgure 1a). In ths examle, two wreless nodes need to exchange ackets x 1 and x through a relay node. A smle store-and-forward aroach needs four transmssons. However, the network codng aroach uses a store-code-andforward aroach n whch the two ackets from the clents are combned by means of an XOR oeraton at the relay and broadcast to both clents smultaneously. The clents can then decode ths coded acket to obtan the ackets they need. Katt et al. [1] resented a ractcal network codng archtecture, referred to as COPE, that mlements the above dea whle also makng use of overheard ackets to ad n decodng. Exermental results shown n [1] ndcate that the network codng technque can result n a sgnfcant mrovement n the network throughut. Research was funded n art by NF grants CN , CMMI , DTRA grant HDTRA , and Qatar Telecom, Doha, Qatar. Fg. 1. a) Wreless Network Codng b) Reverse caroolng. Effros et al. [] ntroduced the strategy of reverse caroolng that allows two nformaton flows travelng n ooste drectons to share a ath. Fgure 1b) shows an examle of two connectons, from n 1 to n 4 and from n 4 to n 1 that share a common ath n 1,n,n 3,n 4 ). The wreless network codng aroach results n a sgnfcant u to 50%) reducton n the number of transmssons for two connectons that use reverse caroolng. In artcular, once the frst connecton s establshed, the second connecton of the same rate) can be establshed n the ooste drecton wth lttle addtonal cost. The key challenge n the desgn of network codng schemes s to maxmze the number of codng oortuntes, where a codng oortunty refers to an event n whch at least one transmsson can be saved by transmttng a combnaton of the ackets. Insuffcent number of codng oortuntes may affect the erformance of a network codng scheme and s one of the major barrers n realzng the codng advantage. Accordngly, the goal of ths aer s to desgn, analyze, and valdate network mechansms and rotocols that mrove the erformance of the network codng schemes through ncreasng the number of codng oortuntes. Consder the scenaro dected n Fgure. We have three sources of traffc, each of whch s aware of two aths leadng to ts destnaton. For examle, ource 3 ostoned at n 5 ) can send ackets to ts destnaton located at n 1 ) at rates x 1 3 and x 3 on ts two avalable aths. Our cost metrc for the system s the number of transmssons er unt tme requred to suort a gven traffc matrx. Under the current channel condtons, suose that t s cheaer for ource 3 to send all ts traffc on ath n 5,n 7,n 1 ). However, notce that there s an oortunty for reverse caroolng on a subath n 1,n,n 5 ). Wth ths scheme, node n wll broadcast coded ackets to nodes n 1 and n 5. We refer to ths broadcast lnk as a hyerlnk. Although ath n 5,n,n 1 ) s more exensve for ource

2 Fg.. Increasng Codng Oortuntes 3, f there s traffc from ource 1 located at n 1 ) that overlas wth t at n, t mght actually be the case that the lowest cost traffc slt n the system would ental that ource3should use the hyer-lnk and send all ts traffc usng ath n 5,n,n 1 ), whle ource 1 follows sut by usng ts ath n 1,n,n 5,n 4 ). However, we mmedately see that there s a rsoners dlemma stuaton here savngs can only be obtaned f there s suffcent b-drectonal traffc on n 1,n,n 5 ). The frst mover n ths case s clearly at a dsadvantage as t essentally creates the route that others can ggyback uon n a reverse drecton). Our challenge n ths aer s the creaton of a dstrbuted control scheme that elmnates the frst-mover dsadvantage, and hence allows the system to attan the state of lowest ossble cost to suort ts traffc. A. Related Work Network codng research was ntated by a semnal aer by Ahlswede et al. [3] and snce then attracted a sgnfcant nterest from the research communty. Many ntal works on the network codng technque focused on establshng multcast connectons between a fxed source and a set of termnal nodes. L et al. [4] showed that the maxmum rate of a multcast connecton s equal to the mnmum caacty of a cut that searates the source and any termnal. In a subsequent work, Koetter and Médard [5] develoed an algebrac framework for network codng and nvestgated lnear network codes for drected grahs wth cycles. Network codng technque for wreless networks has been consdered by Katab et al. [1]. The roosed archtecture, referred to as COPE, contans a secal network codng layer between the IP and MAC layers. In [6] Chachulsk et al. roosed an oortunstc routng rotocol, referred to as MORE, that randomly mxes ackets that belong to the same flow before forwardng them to the next ho. agduyu and Ehremdes [7] focused on the alcatons of network codng n smle ath toologes referred to n [7] as tandem networks) and formulate a related cross-layer otmzaton roblems. mlarly, [8] consders the roblem of utlty maxmzaton when network codng s ossble. However, ther focus s on oortunstc codng as oosed to creatng codng oortuntes that we focus on. Closest to our roblem are [9], [10]. Das et al. [9] roose a new framework called context based routng n multho wreless networks that enables sources to choose routes that ncrease codng oortuntes. They roose a heurstc algorthm that measures the mbalance between flows n ooste drectons, and f ths mbalance s greater than 5%, rovdes a dscount of 5% to the smaller flow. Ths has the effect of ncentvzng equal bdrectonal flows, resultng n multle codng oortuntes. Our objectve s smlar, but we develo terated dstrbuted decson makng that trades off a otental ncrease n cost of longer aths, wth the otental cost reducton due to enhanced codng oortuntes. Marden et al. [10] consder a smlar roblem to ours, but unlke our focus on how to algn user ncentves, ther focus s on the effcency loss of the Nash equlbrum attaned. Our objectve s to desgn an ncentve structure that would naturally result n the system convergng to the lowest cost state. B. Man Results The key contrbuton of ths research s a dstrbuted twolevel control scheme that would teratvely try to lead the sources to dscover the arorate slts for ther traffc among multle aths. On one level are the sources that selfshly choose to slt ther traffc across avalable multle aths wth costs and maxmum caactes set by the hyerlnks) on each. On the other level, the hyerlnk nodes choose maxmum caactes for aths that share that node as a result of the sources decsons as well as oortuntes for network codng. Note by slttng u the dynamcs n ths fashon, our algorthm s a relaxaton of the orgnal costmnmzaton roblem. The teraton rocess contnues untl the entre network has reached local mnmum whch, snce our formulaton s convex, s also the socally otmal soluton. We show that ths rocess s asymtotcally stable. We llustrate our aroach as well as the qualty of soluton usng numercal exerments. The exerments ndcate that: the convergence s fast; the costs are reduced sgnfcantly uon usng network codng; more exensve aths before network codng became cheaer and shortest aths were not necessarly otmal. Thus, the teratve algorthm that we develo from the relaxed formulaton erforms well n ractce. II. YTEM OVERVIEW Our objectve s to desgn a dstrbuted mult-ath network codng system for multle uncast flows traversng a shared wreless network. We assume that the schedule of wreless lnks gven to us for examle, usng CMA), and hence abstract out the nterference between lnks. We model the communcaton network as a grah GV,E), where V s the set of network nodes and E s the set of wreless lnks. For each lnk n,n j ) E, where n,n j ) V, there exsts a wreless channel that allows node n to transmt nformaton to node n j. Each lnk n,n j ) s assocated wth a cost α j. The value of α j catures the cost n number of requred transmssons) of transmttng a acket from n to n j. Due to a broadcast nature of the wreless channels, the node n can transmt to two neghbors n j and n k smultaneously at a cost max{α j,α k }.

3 We assume that the network suorts flows {1,,...,}, where each flow s assocated wth a source and destnaton node. Each flow s also assocated wth several aths {P 1,P,...} that connect ts source and destnaton nodes. Our goal s to buld a dstrbuted traffc management scheme n whch the source node of each flow can slt ts traffc, x ackets er unt tme), among multle dfferent aths, so as to reduce the total number of transmssons er unt tme requred to suort a gven traffc. Note that on some of these aths there mght be a ossblty of network codng. For examle, consder the network dected on Fgure. The network suorts three flows: ) flow 1 fromn 1 to n 4, ) flowfromn 4 ton 6, and ) flow3fromn 5 ton 1. We denote byx the traffc assocated wth flow,1 3. uose that the ackets that belong to flow 1 can be sent over two aths n 1,n,n 3,n 4 ) and n 1,n,n 5,n 4 ). We denote these aths by P1 1 and P1. The traffc slt on aths P1 1 and P1 s gven by x 1 1, x 1, resectvely, such that x1 1 + x 1 = x 1. mlarly, flow can be sent over two aths P 1 = n 4,n 3,n,n 6 ) and P = n 4,n 8,n 6 ) at rates x 1 and x, such that x 1 +x = x. Fnally, flow 3 can be sent over two aths P3 1 = n 5,n 7,n 1 ) and P3 = n 5,n,n 1 ), at rates x 1 3 and x 3, wth sum x 3. Note that ath P1 = n 1,n,n 5,n 4 ) of flow 1 and ath P3 = n 5,n,n 1 ) of flow 3 share two lnks n 1,n ) and n,n 5 ) n the ooste drectons. Thus, the ackets sent along these two aths can beneft from reverse caroolng. ecfcally, node n can combne ackets of flow 1 receved from node n 1 and ackets of flow 3 receved from node n 5. mlarly, node n 3 can combne ackets of flow 1 receved from node n and ackets of flow receved from node n 4. Note that the cost savng at node n s roortonal to mn{x 1,x 3 }, whle the savng at node n 3 s roortonal to mn{x 1 1,x1 }. Note that our model s not restrcted to reverse caroolng tye XOR codng alone. Other tyes of XOR codng schemes lke COPE [11], whch uses oortunstc lstenng can also be used. The cost transmssons er unt tme) at node n when codng s enabled s C n x 1,x 3 ) = max{α 1,α 5 }mn{x 1,x 3 } 1) +α 5 x 1 mn{x 1,x 3}) +α 1 x 3 mn{x 1,x 3 }). Here, the frst term on the rght s the cost ncurred due to codng at node n. Ths s because a coded acket from n s broadcast to both destnaton nodes, n 1 and n 5, and so the cost er acket s max{α 1,α 5 }. The second and thrd term are overflow terms. nce t s ossble that x 1 x 3, the remanng flow of the larger that cannot be encoded because of the lack of flow n the ooste drecton) s sent wthout codng at the regular lnk cost. The cost at node n, gven by 1), can be re-wrtten as shown below: { C n x 1,x 3) = α 5 x 1 +α 1 x 3 + max{α 1,α 5 } } α 1 +α 5 ) mn{x 1,x 3 }. Usng the fact that max{x 1,x } + mn{x 1,x } = x 1 + x, we obtan C n x 1,x 3 ) = α 5x 1 +α 1x 3 ) mn{α 1,α 5 }mn{x 1,x 3}. The above equaton can be nterreted as the cost at node n wthout codng mnus the savngs obtaned when codng s used. Thus, the cost saved at noden due to network codng s mn{α 1,α 5 }mn{x 1,x 3 }. mlarly, for node n 3 the cost saved s mn{α 3,α 34 }mn{x 1 1,x1 }. The total system cost can be exressed as: 3 CX) = β j xj mn{α 1,α 5 }mn{x 1,x 3 } 3) =1 mn{α 3,α 34 }mn{x 1 1,x1 }, where X = {x 1 1,x 1,x1,x,x1 3,x 3 } s the state of the system and β j s the uncoded ath cost equal to the sum of the lnk costs on the ath) j used by flow. For examle, β1 1 = α 1 + α 3 + α 34, for ath P1 1 = n 1,n,n 3,n 4 ). Thus, the frst term on the rght n 3) s the total cost of the system wthout any codng, whle the second and thrd terms are the savngs obtaned by codng at nodes n and n 3. In ths aer, we consder the roblem of mnmzng total cost, gven the traffc matrx. The roblem oses major challenges due to the need to acheve a certan degree of coordnaton among the flows. For examle, for the network dected n Fgure, ncreasng of the value ofx 3 the decson made by node n 5 ) wll result n a system-wde cost reducton only f t s accomaned by the ncrease n the value of x 1. III. HYPER-LINK AND YTEM COT In order to decoule the decsons of flows, we ntroduce the dea of a hyer-lnk whose caacty can be controlled ndeendently of the flows that use t. Defnton 1: A hyer-lnk s a broadcast-lnk comosed of three nodes and two flows. A hyer-lnk n k [,,n ),j,q,n j ))] at node n k can encode ackets belongng to flow sendng ackets on ath ) wth flow j sendng ackets on ath q). Here, nodes n and n j are the next-ho neghbors of n k ; for flow along ath and for flow j along ath q, resectvely. For each hyer-lnk n k [,,n ),j,q,n j ))], we ntroduce a new decson varable y k that denotes the caacty of the hyer-lnk n ackets er unt tme). Ths formulaton hels us to decoule the coordnaton between ndvdual flows. We restrct the total coded broadcast) traffc between the two flows at node n k to be at-most equal to the hyerlnk caacty y k. Any remanng flow s sent wthout codng. Referrng to Fgure, there exsts a hyer-lnk h 1 = n [1,P 1,n 5),3,P 3,n 1)], where the source node n can encode ackets of flow f 1 flow along ath P 1 ), destned to node n 3, wth ackets of flow f 3 flow along ath P 3 ), destned to node n 1. mlarly, there exsts a hyer-lnk h = n 3 [1,P 1 1,n 4),,P 1,n )], where the source node n 3 can encode ackets of flow f 1 1, destned to node n 4, wth

4 ackets of flow f 1 destned to node n. Let the hyer-lnk caactes be defned as y and y 3 resectvely. The total cost of transmsson on hyer-lnk h 1 = n [1,P 1,n 5 ),3,P 3,n 1 )] of caacty y s gven by Ch 1 ) = max{α 5,α 1 }y + 4) α 5 x 1 mn{x 1,y })+ α 1 x 3 mn{x 3,y }) where the frst term on the rght s the cost of sendng traffc on the hyer-lnk. Note that we have to bear ths cost, regardless of whether or not there s enough bdrectonal flow to be sent on the hyer-lnk. Ths relaxaton could otentally ncrease the total cost of the system. However, as we wll see n ecton VI, we can desgn a hyer-lnk caacty controller whch would adjust the hyer-lnk caactes erodcally to mnmze cost. As before, the overflow ackets are sent wthout codng, and the cost ncurred n dong so s gven by the latter two terms. The cost at node n, gven by 4), can be re-wrtten as: Ch 1 ) = α 5 x 1 +α 1 x 3 Th 1 ), where Th 1 ) = α 5 mn{x 1,y }+α 1 mn{x 3,y } max{α 5,α 1 }y Recall that the frst two cost terms are the total cost at node n when codng s dsabled. The remanng cost, Th 1 ), can be thought of as the rebate obtaned by usng hyer-lnk h 1 = n [1,P 1,n 5 ),3,P 3,n 1 )]. Note that the rebate could be negatve hence addng to the total cost), whch mght haen when one of the flow s rate s 0 and the other flow s rate s less than the hyer-lnk caacty. Thus, the modfed cost functon when the system s n state X,Y) s gven by CX,Y) = 3 =1 β j xj Th 1)+Th )), 5) where X = {x 1 1,x 1,x1,x,x1 3,x 3 }, Y = {y,y 3 }. Th 1 ) and Th ) are the rebates obtaned by usng hyer-lnk h 1 = n [1,P 1,n 5 ),3,P 3,n 1 )] and h = n 3 [1,P 1 1,n 4),,P 1,n )]), resectvely. In general, the total system cost n terms of number of transmssons er unt tme requred to suort a gven traffc load, when the state of the system s X,Y), s: CX,Y) = Total system cost wthout codng Total rebate of all the hyer-lnks 6) We focus on mnmzng ths total cost. To ths end, we relax the roblem nto two sub-roblems that of traffc slttng by sources, and that of hyer-lnk caacty selecton: 1) Traffc lttng: In ths hase, the source node of each flow slts ts traffc among the dfferent otons, for a gven hyer-lnk state Y. The otons avalable to each flow are called hyer-aths, where each such hyer-ath contans zero or more hyer-lnks. We model ths hase as a otental game; the background needed s resented n ecton IV. Detals of our game model and the ayoffs used are covered n ecton V. ) Hyer-Lnk Caacty Control: In ths hase, we adjust the hyer-lnk caactes n order to mnmze the total cost. We use a smle gradent descent controller to attan mnmum cost. In ths hase t s assumed the sources attan Wardro equlbrum nstantaneously. Further detals on the tye of controller used and the convergence roertes are covered n ecton VI. We call our controller as Decouled Dynamcs. The two hases oerate at dfferent tme scales. Traffc slttng s done at every small tme scale and the hyer-lnk caacty control s done at every large tme scale. Thus, sources attan equlbrum for gven hyer-lnk caactes, then the hyer-lnk caactes are adjusted, and ths n turn forces the sources to change ther slts. Ths rocess contnues untl the source slts and hyer-lnk caactes converge. IV. BACKGROUND: POTENTIAL GAME Below we revew some game-theoretc deas that wll be used n ths aer. Detaled dscusson may be found n [1]. A oulaton game G, wth F non-atomc oulatons of layers s defned by a mass and a strategy set for each oulaton and a ayoff functon for each strategy. By a nonatomc oulaton, we mean that the contrbuton of each member of the oulaton s nfntesmal. We denote the set of oulatons by F = {1,...,F}, where F 1. The oulaton has mass x. The set of strateges for oulaton s denoted as = {1,..., }. These strateges can be thought of as the actons that members of could ossbly take. A artcular strategy dstrbuton s the way the oulaton arttons tself nto the dfferent actons avalable,.e., a strategy dstrbuton for s a vector of the form x = {x 1,x,...x }, where x = x. The set of strategy dstrbutons of a oulaton F, s denoted by X = { x R + : x = x }. We denote the vector of strategy dstrbutons beng used by the entre oulaton byx = { x 1, x,..., x F }, where x X. The vector X can be thought of as the state of the system. Let the sace of all strategy dstrbutons be X. The margnal ayoff functon er unt mass) obtaned from strategy by users of class, when the state of the system s X s denoted by F X) R and s assumed to be contnuous and dfferentable. Note that the ayoffs to a strategy n oulatoncan deend on the strategy dstrbuton wthn oulaton tself. The total ayoff to users of class s then gven by F X)x, where we assume lnearty for exoston. Potental games are a tye of oulaton games, that have a secfc structure on the cost functon. The dea behnd otental games s to dentfy a scalar functon that reresents the energy of the system exactly lke a Lyaunov functon [13]), whch s called the otental functon. All nformaton regardng the ayoffs obtaned by users of a oulaton class can be catured n the otental functon.

5 Defnton : Let G be a oulaton game wth ayoff functon er unt mass) F : X R F. G s called a Potental Game f there exsts a contnuously dfferentable functon T : X R such that T x X) = F X) 7) F and, where X X s the state of the system. The functon T s called the otental functon for game G. Next, we defne the concet of equlbrum n oulaton games. A commonly used concet n non-cooeratve games n the context of nfntesmal layers, s the Wardro equlbrum [14]. Consder any strategy dstrbuton x = [x 1,...,x ]. There would be some elements whch are non-zero and others whch are zero. We call the strateges corresondng to the non-zero elements as the strateges used by oulaton. Defnton 3: A state ˆX s a Wardro equlbrum f for any oulaton F, all strateges beng used by the members of yeld the same margnal ayoff to each member of, whereas the margnal ayoff that would be obtaned s lower for all strateges not used by oulaton. Let Ŝ be the set of all strateges used by oulaton n a strategy dstrbuton ˆX. A Wardro equlbrum ˆX s then characterzed by the followng relaton: F s ˆX) F s ˆX) s Ŝ and s The above concet refers to an equlbrum condton; the queston arses as to how the system actually arrves at such a state. A commonly used knd of oulaton dynamcs s Brown-von Neumann-Nash BNN) Dynamcs [15]. The dynamcs are descrbed as follows: ẋ = x γ x γ j 8) where, γ = max F 1 x F j xj,0 Note that the total mass of the oulaton s a constant x. An nterestng roerty of BNN dynamcs s non-comlacency,.e., t allows extnct strateges to resurface, so that ts statonary onts are always Wardro equlbra [1]. V. TRAFFIC PLITTING: MULTI-PATH NETWORK CODING MPNC) GAME We model the traffc-slttng rocess of our Decouled Dynamcs controller as a otental game, G, whch we refer to as the Mult-Path Network Codng Game MPNC Game). Our system model conssts of a set of nodes N = {n 1,...,n N }, where each node n N s surrounded by a random number of other nodes. The cost of transmsson er acket) from node n to ts neghborng node n j s a constant and s equal to α j, smlarly, cost of transmsson er acket) from n j to n s α j. There exsts a set of flows these corresond to layers n the game) F = {1,...,F}. Each flow, F s defned as a tule n s,nd,x ), where n s N s the source node,n d N s the destnaton node, andx ackets/sec s the traffc sent from source to destnaton. Ths traffc s equvalent to the oulaton mass n the oulaton game nterretaton. Each flow s assocated wth a set of hyer-aths. Defnton 4: A hyer-ath between source n s and destnaton n d s a vrtual ath over a hyscal ath between n s and nd. A hyer-ath contans zero or more hyer-lnks on t and at each node on the underlyng hyscal ath there can be at-most one hyer-lnk. It follows that the set of all aths are a subset of the hyer-aths. In other words, a hyer-ath can have a combnaton of at-most two flows at each node. A flow can slt ts traffc among the hyer-aths avalable to t, and we denote a sub-flow f of flow by the tule n s,nd,,x ). Here, x s the traffc sent by flow on hyer-ath. The sum of lnk costs er unt rate) on the hyscal ath corresondng to the hyer-ath s denotedβ. Note that the cost seen by a sub-flow usng such a hyer-ath mght be lower than ths cost due to savng attaned by network codng. We reresent the dvson of traffc x of flow F, over all the hyer-aths as a vector, x = {x 1,...,x } such that x = x. x s called the strategy dstrbuton of flow, and the set of all the strategy dstrbutons of all the flows s called the state of the flows and s reresented as X = [ x 1... x F ]. We denote the set of all states of the system as X,.e., X X. The set of all hyer-lnks n the network s assumed to be re-determned and s reresented by H = {1,...,H}, where H s the number of hyer-lnks. Recall that the hyer-lnk formed by encodng ackets that belong to flows and j, for,j F at node n k s reresented by n k [,,n ),j,q,n j )]. Nodes n and n j are the next ho nodes for the hyer-ath of flows and j, usng hyer-aths and q resectvely. Note that we have slghtly modfed the defnton to nclude the fact that and j are usng hyer-aths. We denote by H H the set of all hyer-lnks assocated wth flow f. Each hyer-lnk can choose ts caacty ndeendently of others. We denote the caacty of the hyer-lnk h = n k [,,n ),j,q,n j )] by y h ackets/sec. The hyer-lnk broadcasts ackets receved at node n k to n and n j u to caacty y h. The vector of all hyer-lnk caactes s called the hyer-lnk state and s denoted by, Y = [y 1,...,y H ]. Let Y be the set of all ossble hyer-lnk states,.e., Y Y. The state of the system s defned as X,Y), where X X s the state of the flows and Y Y s the state of the hyer-lnks. In the traffc slttng hase of our algorthm, flows try to attan the state of lowest cost for a gven hyer-lnk state Y. The hyer-lnk caactes are controlled n the next hase hyer-lnk caacty control), dscussed n ecton VI The ayoff er unt rate) obtaned n usng hyer-ath of flow F when the state of the system s X,Y) s denoted by F X,Y) R and s assumed to be contnuous and dfferentable. We may have to make sutable aroxmatons on cost functons to ensure that these condtons hold. We model our system as a otental game, usng the total cost functon CX, Y) as our otental functon.

6 Recall from 6) that the total cost of the system s where CX,Y) = F =1 β x H h=1th), 9) Th) = α k mn{x,y h}+α kj mn{x q j,y h} max{α k,α kj }y h 10) As can be seen from 10), the cost functon contans mn terms over the hyer-lnk caacty and the flow rates, ths makes the functon non-contnuous and non-dfferentable. In order to have a contnuously dfferentable cost functon we aroxmate these mn terms usng a generalzed meanvalued functon. Let a = {a 1,...,a n } be the set of ostve real numbers and let r be some non-zero real number. Then the generalzed r-mean of a s gven by: ) 1 n M r a) = a r 11) n =1 The mn functon over the set a s aroxmated usng M r a) as: mn{a 1,...,a n } = lm r M ra) 1) ubsttutng for M r 11), nstead of the mn functon n 9) we get the aroxmated total cost functon as: CX,Y) = F =1 β x H Th), h=1 13) where for a hyer-lnk h = n k [,,n ),j,q,n j )] H: x ) 1 r x ) Th) = α k +α )r +y h ) r 1 r kj )r +y h ) r max{α k,α kj }y h 14) The cost functon CX,Y) s contnuous and dfferentable. o, we use the aroxmated cost functon as our otental functon. Thus, t follows from the defnton of otental games Defnton ) that, the ayoff obtaned by flow F n usng oton s: where, from 14) F CX,Y) X,Y) = x Th) x = α k Recall that M r x x,y h) = Hence, = β h H F X,Y) = β h H F, 15) Th), 16) x x ) r 1 M r x,y 17) h) )r +y h ) r α k ) 1 r. 18) x r 1, M rx h)) 19),y where H s the set of all hyer-lnks assocated wth sub-flow. Note, the ayoff s the cost ncurred n usng an oton, f so the layers try to mnmze ther cost. The source node of each flow, F, observes the margnal cost, F, obtaned n usng a artcular oton,, and changes the mass on that artcular oton,x, so as to attan Wardro equlbrum [14]. The source nodes use BNN dynamcs 8) to control the mass on each oton. But snce each source tres to mnmze ts ayoff, we use a modfed verson BNN dynamcs: ẋ f = γ f f where, γ f = max 1 F j f xj f F f,0 γ j f, 0) In the next secton, we rove the stablty of our system usng Lyaunov theory. A. Convergence of MPNC Game We show n ths secton that the mult-ath network codng game converges to a statonary ont when each source uses BNN dynamcs. We wll use the theory of Lyaunov functons [13] to show that our oulaton game G, s stable for a gven hyer-lnk state Ŷ. We use the aroxmated total cost of the system 13) as our Lyaunov functon. Theorem 1: The system of flows F that use BNN dynamcs wth ayoffs gven by 19) s globally asymtotcally stable for a gven hyer-lnk state Ŷ. Proof: We use the aroxmated total cost functon CX,Y) 13) as our Lyaunov functon. It s smle to verfy that the cost functon CX,Ŷ), s non-negatve and convex, and hence s a vald canddate. For a gven hyer-lnk state, Ŷ, we defne our Lyaunov functon as: From 15) Hence, LŶX) x f LŶX) = = CX,Ŷ) x f LŶX) = F = F From 0) we can substtute the value for ẋ f = F LŶX) = CX,Ŷ) 1) = F f X,Ŷ). ) LŶ X) ẋ x f 3) f F f X,Ŷ)ẋ f 4) and we have F F f x fγ f γ j f ) Ff P γ f 1 f F f γ j f 5)

7 We defne Thus, f F f 1 F f x f F = Ff P γ f F f γ j f 6) = F F γ f FP f F f ) 7) γ f ) 0 8) LŶX) 0, X X 9) where equalty exsts when the state X corresonds to the statonary ont of BNN dynamcs. Hence, the system s globally asymtotcally stable. B. Effcency The objectve of our system s to mnmze the total cost for a gven load vector x = [x 1,...,x Q ] and gven hyerlnk state Ŷ. Here the total cost n the system s CX,Ŷ) and s defned n 9). Ths can be reresented as the followng constraned mnmzaton roblem: mn X CX,Ŷ) 30) subject to: x = x F 31) x 0. The Lagrange dual assocated wth the above mnmzaton roblem, for a gven Ŷ s LŶλ,h,X) = max mn CX,Ŷ) 3) λ,h X F ) F λ x x ) =1 =1 h x where λ and h 0, F and, are the dual varables. Now the above dual roblem gves the followng Karush-Kuhn-Tucker frst order condtons: LŶ x λ,h,x ) = 0 F and 33) and h x = 0 F and 34) where X s the global mnmum for the rmal roblem 30). Hence from 33) we have, F and, C x X,Ŷ) λ x x ) x +h = 0 C x X,Ŷ) = λ +h 35) F X,Ŷ) = λ +h 36) where the last equaton follows from Defnton. From 34), t follows that F X,Ŷ) = λ and F X,Ŷ) = λ +h when x > 0 37) when x = 0 38) F and. The above condton 37, 38), mles that the ayoff on all the otons used s dentcal and for otons not n use the ayoff s more, whch s equvalent to the defnton of Wardro equlbrum Defnton 3). Notce, we use a modfed defnton of Wardro equlbrum, snce each source tres to mnmze t s cost or ayoff). The followng theorem roves the effcency of our system. Theorem : The soluton of the mnmzaton roblem n 30) s dentcal to the Wardro equlbrum of the noncooeratve otental game G. Proof: Consder the BNN dynamcs 0), at statonary ont, X, we have ẋ = 0, whch mles that ether, ˆF = F X,Ŷ) 39) or ˆx = 0, where, ˆF 1ˆx Q r=1 ˆxr Fr X,Ŷ) F, 40) The above exressons mly that all hyer-aths used by a artcular flow F yeld same ayoff, ˆF, whle hyer-aths not used x = 0) yeld a ayoff hgher than ˆF. We observe that the condtons requred for Wardro equlbrum are dentcal to the KKT frst order condtons 37)- 38) of the mnmzaton roblem 30) when ˆF = λ F It follows from the convexty of the total system cost that, there s no dualty-ga between the rmal 30) and the dual 3) roblems. Thus, the otmal rmal soluton s equal to otmal dual solutons, whch s dentcal to the Wardro equlbrum. VI. HYPER-LINK CAPACITY CONTROL Thus far we have desgned a dstrbuted scheme that would result n mnmum cost for a gven hyer-lnk state or caactes Y and for a gven load vector x = {x 1,..., }. In ths hase of Decouled Dynamcs, the hyer-lnk caactes are adjusted based on the current system cost so as to guarantee a mnmum total system cost for a gven load vector x. Ths hase runs at a larger tme-scale as comared to the traffc slttng hase descrbed n ecton V. It s assumed that durng ths hase all the flows or layers reman n equlbrum,

8 .e., changng the hyer-lnk caactes would force all the source nodes to attan Wardro equlbrum nstantaneously. The hyer-lnk caacty control can be formulated as a centralzed convex otmzaton roblem as follows: mn Y subject to, HY) 41) y h 0 y h Y and h H where, HY) s the mnmum total cost of the system for a gven hyer-lnk state Y,.e., HY) = CX,Y), where, for a gven Y, X s an otmal state of the flows that results n mnmum cost. 1 We use smle gradent descent: ẏ h = κ HY) y h Y 4) The artal dervatve, HY ), s over the varables y h Y. Changng the hyer-lnk caacty y h, of some hyer-lnk h H, would result n a dfferent state of the flows, X h and hence a dfferent mnmum cost, CX h,y h), where Y h corresonds to the changed hyer-lnk caacty of y h whle other caactes are fxed, as comared to Y. Thus for a hyerlnk, h = n k [f,n ),f q j,n j)] wth caacty y h, HY) = C X,Y)+ F =1 C X,Y) x x = C X,Y)+ F =1 F x 43) where, the last exresson follows from the defnton of F Defnton 15) and the fact that, for changes n the hyer-lnk state, the sources attan Wardro equlbrum nstantaneously. In other words, before and after a small change n y h the system s n Wardro equlbrum. Hence, F = F F and. where, the last exresson follows from the defnton of F Defnton 15) and the fact that, for changes n the hyer-lnk state, the sources attan Wardro equlbrum nstantaneously. In other words, before and after a small change n y h the system s n Wardro equlbrum. Hence, F = F F and. Fnally, the total load x = x s fxed. HY) x = 0 snce, = C X,Y) = T h) 44) where, from 14), for hyer-lnk h = n k [f,n ),f q j,n j)], T h) = α k y h y h M rx,y h) Recall, M r x,y h) = ) r 1 + α kj max{α k,α kj } ) r 1 y h M rx q j,y h) x ) )r +y h ) r 1 r Theorem 3: At the large tme-scale, the hyer-lnk caacty control wth dynamcs 44) s globally asymtotcally stable. Proof: We use the followng Lyaunov functon ZY) = VY) VŶ) 45) where VY) = κhy) 46) 1 Notce, there could be many dfferent states, X, whch result n a mnmum cost but the mnmum value, CX,Y), s unque. whch s strctly convex, wth Ŷ s the hyer-lnk state whch results n mnmum cost HY). Dfferentatng ZY) we obtan H V Ż = y h. 47) Then from 46) and 44) h=1 V = κ HY) ẏ h = ẏh yh 48) Ż = F h=1 ẏ h y h 0 Y, 49) wth Ż = 0 at the statonary onts of the system. Thus, the system s globally asymtotcally stable [13]. Fnally, t s not hard to show that the equlbrum condtons of the controller 44) are the same as the KKT condtons of the otmzaton roblem 41). Hence, the controller succeeds n mnmzng the total cost of the system for a gven load x nto the system. Thus, the system state converges to a local mnmum. Now, snce the global cost mnmzaton roblem under the mn aroxmaton s convex, the soluton s the global mnmum of the relaxed roblem. VII. IMULATION We smulated our system n Matlab to show system convergence. We frst erformed our smulatons for our smle network shown n Fgure. The load at the source nodes 1, and 3 s gven as 4.73,.69 and 3.56 resectvely, whch are randomly generated values. We use the followng costs on the ndvdual lnks α j ): α 1 =.8, α 3 = 1.6, α 34 = 1.8, α 5 = 1.3, α 54 =.1, α 6 = 1.7, α 48 =.9, α 86 =., α 57 = 1.9, α 71 =.6; we assume the costs on the lnks are symmetrc. We use the aroxmated cost functon 13), wth a value of r = 100 for the aroxmaton arameter 1) for our smulatons. The smulaton s run for 50 large tme unts, and n each large tme scale we have 0 small tme unts. We comare the total cost of the system for the followng: 1) Decouled Dynamcs DD): Ths s the algorthm that we develoed; we use our hyer-lnks to decoule the flows that artcate n codng. ) Couled Dynamcs no hyer-lnk) CD): Here, there s coulng between ndvdual flows and codng haens at the mnmum rate of the consttuent flows. We use smlar game dynamcs as that was used n DD. The total cost s secfed n Equaton 3). 3) No Codng: In ths system no network codng s used. 4) LP Otmal LP): Ths s a centralzed soluton. We formulated our system as a Lnear Program LP) of mnmzng cost 9) over X and Y for a gven load vector that we obtan usng an LP-solver. As seen n the Fgure 3, the total cost of the system number of transmssons er unt tme) for our model decouled usng hyer-lnk) s close to the otmal soluton obtaned by solvng t n a centralzed fashon. We comared the fnal system state of DD and CD wth that of the soluton obtaned usng LP. We observe from Table I that the values for the slt X) and the

9 60 59 Total ystem Cost Decouled hyer lnk) Couled no hyer lnk) No network Codng Lnear Program otmal) Tme Large Tme cale) Fg. 3. Comarson of total system cost er unt rate), for dfferent systems: DD, CD and non-coded aganst LP. Fg. 4. Network toology. Varable x 1 1 x 1 x 1 x x 1 3 x 3 y y 3 LP DD CD NA NA TABLE I COMPARION OF TATE VARIABLE FOR LP AND DD AND CD. hyer-lnk caactes Y ) generated by DD are near-otmal LP results), but CD s very dfferent. Next, we erform our smulatons on a bgger toology shown n Fgure 4. Ths network conssts of 30 nodes shared by 6 flows.flows 1,, 3 and 6 have two hyer-aths each and flows 4 and 5 have three hyer-aths each. There are 6 hyerlnks n the system. We ran our algorthms on ths network wth random lnk costs, and show the results for four examle cases n Table II. We observe that DD erforms near-otmally and sgnfcantly outerforms CD n terms of total cost. VIII. CONCLUION We consder a wreless network wth gven costs on arcs, traffc matrx and multle aths. The objectve s to fnd the slts of traffc for each source across ts multle aths n a dstrbuted manner leveragng the reverse caroolng technque. For ths we relax the roblem nto two sub-roblems, and roose a two-level dstrbuted control scheme set u as a game between the sources and the hyerlnk nodes. On one level, gven a set of hyerlnk caactes, the sources selfshly choose ther slts and attan a Wardro equlbrum. On the other level, gven the traffc slts, the hyerlnks may slghtly ncrease or decrease ther caactes usng a steeest descent algorthm. We construct a Lyaunov functon argument to show that ths rocess asymtotcally converges, although erformed selfshly n a dstrbuted fashon. We erformed several numercal studes and found that our two-level controller converges fast to the otmal solutons. case LP DD CD TABLE II TOTAL YTEM COT COMPARION OF DECOUPLED DYNAMICDD) AND COUPLED DYNAMIC CD) AGAINT THE LP OLUTION. ome of the b-roducts of our exerments were that: more exensve aths before network codng became cheaer and shortest aths were not necessarly otmal. In concluson, from a methodologcal standont we have a dstrbuted controller that acheves a near-otmal soluton when the ndvduals are self-nterested. REFERENCE [1]. Katt, H. Rahul, D. Katab, W. H. M. Médard, and J. Crowcroft, XORs n the Ar: Practcal Wreless Network Codng, n ACM IG- COMM, Psa, Italy, 006. [] M. Effros, T. Ho, and. Km, A tlng aroach to network code desgn for wreless networks, Informaton Theory Worksho, 006. ITW 06 Punta del Este. IEEE,. 6 66, March 006. [3] R. Ahlswede, N. Ca,.-Y. R. L, and R. W. Yeung, Network Informaton Flow, IEEE Transactons on Informaton Theory, vol. 46, no. 4, , 000. [4].-Y. R. L, R. W. Yeung, and N. Ca, Lnear Network Codng, IEEE Transactons on Informaton Theory, vol. 49, no., , 003. [5] R. Koetter and M. Medard, An Algebrac Aroach to Network Codng, IEEE/ACM Transactons on Networkng, vol. 11, no. 5, , 003. [6]. Chachulsk, M. Jennngs,. Katt, and D. Katab, Tradng structure for randomness n wreless oortunstc routng, n IGCOMM 07: Proceedngs of the 007 conference on Alcatons, technologes, archtectures, and rotocols for comuter communcatons. New York, NY, UA: ACM, 007, [7] Y. agduyu and A. Ehremdes, Cross-layer otmzaton of MAC and network codng n wreless queueng tandem networks, Informaton Theory, IEEE Transactons on, vol. 54, no., , Feb [8] U. K. Hulya eferogluy, Athna Markooulouy, Network codng-aware rate control and schedulng n wreless networks, n Proc. IEEE Internatonal Conference on Multmeda and Exo, Jun 009. [9]. Das, Y. Wu, R. Chandra, and Y. Hu, Context-based routng: technques, alcatons and exerence, n Proceedngs of the 5th UENIX ymosum on Networked ystems Desgn and Imlementaton table of contents. UENIX Assocaton Berkeley, CA, UA, 008, [10] J. Marden and M. Effros, The Prce of elfshness n Network Codng, n Worksho on Network Codng, Theory, and Alcatons, 009. [11]. Katt, H. Rahul, W. Hu, D. Katab, M. Médard, and J. Crowcroft, Xors n the ar: ractcal wreless network codng, IGCOMM Comut. Commun. Rev., vol. 36, no. 4, , 006. [1] W. H. andholm, Potental Games wth Contnuous Player ets, Journal of Economc Theory, vol. 97, , January 001. [13] H. Khall, Nonlnear ystems. Prentce Hall, [14] J. G. Wardro, ome theoretcal asects of road traffc research, n Proc. of the Insttute of Cvl Engneers, vol. 1, 195, [15] G. W. Brown and J. von Neumann, oluton of games by dfferental equatons, Contrbutons to the Theory of Games I, Annals of Mathematcal tudes, vol. 4, 1950.