A Cro-Layer Optimization Framework for Muticat in Muti-hop Wiree Network Jun Yuan, Zongpeng Li, Wei Yu, Baochun Li Department of Eectrica and Computer Engineering Univerity of Toronto {teveyuan@comm, arcane@eecg, weiyu@comm, bi@eecg}.toronto.edu Abtract Achieving optima tranmiion throughput in data network i known a a fundamenta but hard probem. The ituation i exacerbated in muti-hop wiree network due to the interference among oca wiree tranmiion. In thi paper, we propoe a genera modeing and oution framework for the throughput optimization probem in wiree network. In our framework, data routing, wiree medium contention and network coding are jointy conidered to achieve the optima network performance. The prima-dua oution method in the framework repreent a cro-ayer optimization approach. It decompoe the origina probem into data routing ub-probem at the network ayer, and power aocation ub-probem at the phyica ayer. Variou effective oution are dicued for each ub-probem, verifying that our framework may hande the throughput optimization probem in an efficient and ditributed fahion for a broad range of wiree network cenario. I. INTRODUCTION Muti-hop wiree network conit of wiree node that communicate with each other by reaying data fow over mutipe hop, without infratructure upport. Due to the broadcat nature of omni-directiona antenna, geographicay nearby tranmiion interfere with each other. In thi paper, we tudy the probem of achieving optima tranmiion throughput for mutipe concurrent data eion in muti-hop wiree network. A unicat, broadcat and group communication eion may a be viewed a (or tranformed into) muticat eion [1], we aume that a data eion i a muticat eion without o of generaity. A deirabe oution to the probem of achieving optima tranmiion throughput incude a routing trategy of data fow at the network ayer, a we a a power aocation cheme that ead to high capacity at the phyica ayer. A the main origina contribution of thi paper, we propoe a genera framework to mode and ove the optima throughput probem in wiree network. At a high eve, our framework imize the overa throughput ubject to three group of contraint: (1) dependence of overa throughput on per-ink data fow rate, () dependence of per-ink fow rate on ink capacitie, and (3) dependence of ink capacitie on pernode radio power eve. We preent a genera prima-dua method that iterativey ove two dijoint ub-probem and converge to the optima oution of the origina probem. The firt ub-probem i a muti-hop fow routing probem at the network ayer, and the econd ub-probem i a power aocation probem at the phyica ayer. We further iutrate how each ub-probem can be oved efficienty under different wiree network cenario. We ao take different agorithmic perpective: at the network ayer, we dicu the probem with or without network coding; at the phyica ayer, we conider the dua optimization method, geometric programming, a we a game theoretic deign. The genera framework propoed in thi paper repreent a cro-ayer optimization trategy. It baance the demand and uppy of ink bandwidth at the network and the phyica ayer, repectivey. It ao provide an optima fow routing cheme and a correponding optima power aocation cheme in uch a baanced tate. Important propertie and highight of our oution framework are a foow. Comprehenivene: It provide a comprehenive approach for ytem deigner to incorporate networking factor from different ayer into a joint optimization probem. Such joint conideration of network coding, data routing, and wiree interference i indeed neceary to approach optima performance in muti-hop wiree network. Divide and conquer trategie: It expore the underying moduar tructure of the joint optimization probem from the perpective of prima-dua oution. It appie the caic technique, Lagrange reaxation combined with ubgradient optimization, to decompoe the origina compex probem into maer ub-probem that are eaier to ove. Each ub-probem reide in ony one networking ayer, it therefore provide a cear jutification for ayered protoco deign. Fexibiity: How each ub-probem i oved depend on the pecific mode and the avaiabe oution technique, and i independent of the genera framework that we propoe. A new advance occur in networking technoogie or in optimization agorithm deign, the modue for oving the correponding ub-probem in our framework can be eaiy repaced to refect uch advance and to achieve better performance. A exampe, the modue for fow routing may be impemented a muticommodity fow routing, tree packing and network coding; and the modue for power aocation can be computed with the dua optimization method, geometrica programming, and a cooperative efih game. Effectivene: Our oution framework tranform the overa throughput optimization into a equence of imper ub-probem. A ong a the computation of each
individua ub-probem i optima, efficient and/or ditributed, the overa oution i guaranteed to be optima, efficient, and/or ditributed, repectivey. We ha iutrate how optimaity can be achieved or coey approached for each ub-probem, in an efficient and ditributed fahion. The remainder of thi paper i organized a foow. We firt dicu reated work in Section II. In Section III, we propoe the joint optimization framework and the ayering approach, together with an efficient prima-dua agorithm to ove the probem. In Section IV, we dicu the modue tructure of ub-probem, point out evera new technique in network and phyica ayer, and how how the ub-probem are incorporated in the overa framework. We then preent an exampe to iutrate the main concept in Section V. Finay, we dicu the imitation and extenion of thi paper in Section VI, and concude in Section VII. II. RELATED WORK In the genera mode of data network, recent reearch in information theory dicover that routing aone i not ufficient to achieve imum information tranmiion rate [], [3]. Rather, encoding and decoding operation at reay node in addition to the ender and receiver are in genera neceary in an optima tranmiion trategy. Such coding operation are referred to a network coding. The pioneering work by Ahwede, Cai, Li and Yeung [] and Koetter and Medard [3] prove that, in a directed network with network coding, a muticat rate i feaibe if and ony if it i feaibe for a unicat from the ender to each receiver. Li, Yeung and Cai [4] then how that inear coding uuay uffice in achieving the imum rate. With the aitance of network coding, the probem of achieving optima throughput ha been tudied in undirected network where each ink ha a known capacity, hared by fow in both direction. A hown by Li et a [1], [5], the probem of computing optima throughput can be formuated a a inear optimization probem, and ditributed agorithm can be deigned to efficienty ove the probem by appying Lagrange reaxation and ubgradient optimization. However, the aumption of fixed ink capacitie i not reaitic in muti-hop wiree network, where ink capacitie are ubject to interference from other ink in the neighborhood. If we aume that two tranmiion can be imutaneouy active if and ony if neither node in one tranmiion i within the communication range of the other tranmiion [6] (ao referred to a the ogica interference mode), then even optimizing ink cheduing for a et of fixed muti-hop fow i NP-hard, ince it eentiay correpond to the graph cooring probem. A main contribution of thi paper i to take the phyica ayer interference into account when oving the optima throughput probem for muti-hop wiree network. Toward thi objective, we draw from previou tudie reated to power contro in wiree network and how how phyica ayer mode can be incorporated in the overa probem of optimizing throughput, with the aitance of network coding. In particuar, we take advantage of recent progre in oving non-convex probem uing a dua optimization method a propoed by Yu et a [7] for Orthogona Frequency Diviion Mutipex (OFDM) ytem in which the non-convex probem i oved efficienty and gobay in the dua domain, and the geometric programming approach propoed by Chiang [8] for Code Diviion Mutipe Acce (CDMA) ytem in which the non-convex probem i tranformed into a convex one under the aumption of a high igna to interference and noie ratio (SINR). Finay, we ao propoe an approximate but nearoptima oution for the interference channe baed on game theory. The main technique ued in thi paper i the method of dua decompoition for convex optimization probem. The dua decompoition technique i reated to the duaity anayi of TCP a a fow contro protoco by Low [9] and Wang et a [10], in which network congetion parameter are interpreted a prima and dua optimization variabe and the TCP protoco i interpreted a a ditributed prima-dua agorithm. Our work i ao reated to the extenion of the above work to mutihop wiree network by Chiang [8], in which power eve and TCP window ize are jointy optimized, and a dua variabe i ued a a mean for cro-ayer optimization. In a reated work, Johanon, Xiao, and Boyd have ao carried out a imiar convex optimization approach to jointy perform routing and reource aocation in wiree CDMA network [11], [1], where a high SINR i aumed in order to guarantee convexity. In our previou work [13], we have ao tudied a dua method for the joint ource coding, routing, and power aocation probem for enor network, where the focu i a oy ource coding probem in the appication ayer. A of the above work treat the muti-eion unicat probem ony. The main idea of the preent work i to propoe a imiar framework for muticat probem in a network coding context. For wiree muticat in ad hoc network, Wu et a [14] tudied the iue of network panning and oved an energy minimization probem with centraized contro. They ao adopt SINR to mode contention at the phyica ayer, and their tranmiion pan invove a time haring cheme among a et of eected power aocation tate. Both the focu and oution approache of our paper are different a compared to [14]. We target imum throughput intead of minimum cot, and preent a genera oution framework that decompoe the optimization into different ayer. We ao target ditributed oution and empoy phyica ayer mode, which directy connect capacity rate to power aocation. III. A JOINT OPTIMIZATION FRAMEWORK We now preent a genera framework to mode and ove the probem of optimizing throughput in muti-hop wiree network. We firt give a high-eve formuation of the optimization probem, which invove variabe from both the network ayer and the phyica ayer. We then how that Lagrange reaxation and ubgradient optimization can be appied to decompoe uch an overa optimization probem into a equence of maer ub-probem, each ony invoving variabe from either the network ayer or the phyica ayer. Interaction between the two ub-probem are then dicued.
A. Genera Framework of Joint Optimization The formuation of the throughput optimization probem i baed on the foowing fact. Firt, throughput i reaized by routing data fow from ender to receiver, therefore the achievabe throughput for each data eion i decided by the correponding data fow of that eion. Second, at each tranmiion ink, the aggregated data fow rate can t exceed the effective capacity of that ink. Third, the achievabe ink capacity i decided by SINR, which in turn i decided by the power eve at a the ender. Let G = (V, E) be the network topoogy. Let S be the et of mutipe data eion upported in the network. Let r = {r i } be the et of muticat rate for each eion i S. Fix r, et N (r) be the et of network fow rate that are needed to upport r. The fow rate f i a vector {f i }, where i denote the eion index i S and denote the ink index E. Let p be the et of power contraint on each node in V. Let C(p ) be the et of achievabe rate c = {c } that the phyica ayer can upport on each ink E. The throughput optimization probem can now be formuated a: U(r) (1).t. f N (r) c C(p ) f i c, i S E where we eek to imize ome concave utiity function of the throughput vector U(r). For exampe, a utiity that ead to proportiona fairne i U(r) = i U i(r i ) = i og(ri ). The contraint f N (r) mode the inter-dependence of achievabe throughput r and the data fow routing cheme f. The contraint c C(p ) mode the inter-dependence of ink capacity vector c on the node power contraint p. The contraint i f i c refect the fact that the aggregated fow rate at each ink i bounded by ink capacity. Here, i i the index of data eion, and i the index of ink. The detaied characterization of the region N (r) and C(p ) are independent of our genera formuation and wi be dicued in the next ection. We now proceed to invetigate oution technique that are appicabe to our genera probem formuation. B. Decompoing the Probem When both N (r) and C(p ) are convex region, genera convex optimization method can be ued to ove the overa optimization probem (1). However, uch a oution doe not take advantage of the pecia probem tructure and it in genera require goba information to be coected at a centra point of computation. In thi paper, we intead propoe a genera oution framework, within which the origina probem i decompoed into maer ub-probem, each of which can be oved efficienty and ditributivey. We tart by reaxing the ink capacity contraint i f i c and introduce price into the objective function: L = U(r) + [ λ c i f i ]. () Oberve that the imization of the Lagrangian above now conit of two et of variabe: network ayer variabe (f, r), and phyica ayer variabe (p, c), where p i a et of ink power contrained by node power budget p. More pecificay, the Lagrangian optimization probem now decoupe into two dijoint part. The network ayer part i a data fow routing probem: U(r) λ f i (3) i.t. f N (r) The phyica ayer part i a power aocation probem:.t. c C(p ) λ c (4) Thu, the optimization framework naturay provide a ayering approach to the wiree throughput optimization probem. The goba imization probem decompoe into two part: routing at the network ayer and power aocation at the phyica ayer. The power aocation probem enure that the ima capacity i provided in individua network ink, whie the routing probem enure that underying ink upport i efficienty utiized to imize the muticat rate. The decouping of the network optimization probem ao revea that cro-ayer deign can be achieved in a theoreticay optima way. The dua variabe λ pay a key roe in coordinating the network ayer demand and phyica ayer uppy. In particuar, the th component of λ (λ ) can be interpreted a the rate cot in ink. A higher vaue of λ igna to the underying phyica ayer that more reource houd be devoted to tranporting the traffic in ink. At the ame time, it igna to the upper network ayer that tranporting bit in ink i expenive and it provide incentive for the network ayer to find aternative route for traffic. The key requirement that aow the decouping of the network optimization probem into routing and reource aocation i the underying convexity tructure of the probem. Beow, we firt provide a jutification for convexity, then propoe a prima-dua agorithm that can be ued to ove the joint network optimization probem efficienty. C. The Roe of Convexity We tart by oberving that at the phyica ayer, C(p ) can away be made a convex region and C(p ) can away be made a convex function of p via time haring. If two et of rate are both achievabe under the ame power contraint, then their inear combination mut ao be achievabe by impy dividing the frequency (or time) into two ub-channe and tranmitting uing the two different trategie in the two ub-channe. From a network routing point of view, the technique of network coding reove the competition among fow by
introducing conceptua fow [1]. The network coded routing region pecifie three kind of inear contraint for conceptua fow: (a) the imum fow rate mut be upper bounded by the ink capacity; (b) the aw of fow conervation, i.e., the incoming conceptua fow rate i equa to the outgoing conceptua fow rate at a reay node for every given ourceink pair; and (c) the muticat rate mut be e than or equa to the rate for each ource-ink pair in every eion. Hence, it i obviou that the contraint et of network coded routing region i convex. D. The Prima-Dua Soution Framework A direct impication of the convexity of the capacity region C(p ) and the data routing region N (r) i that the joint network optimization probem (1) can be oved efficienty. Further, a trong duaity hod, the optimization probem (1) can be oved via it dua. A the dua probem ha a natura decompoition, it may be oved by oving it two ubprobem (3) and (4) individuay and by updating the hadow price λ in ucceion. More pecificay, we now propoe the foowing primadua agorithm that ove the entire network optimization probem: Agorithm 1: Prima-Dua Agorithm: 1) Initiaize λ (0) ) In prima domain, given λ (t), ove the foowing ubprobem: { U(r) } λ f i f N (r) (5) f, r i { } λ c c C(p ) (6) p, c 3) In dua domain, update λ uing the foowing rue: λ (t+1) = λ (t) + ν (t) (c i 4) Return to tep unti convergence. f i ) (7) Theorem 1: Agorithm 1 away converge to the goba optimum of the overa network optimization probem (1), provided that the tep ize ν (t) i choen to be ufficienty ma. Proof: We outine the proof here. The crucia ingredient that vaidate Agorithm 1 i convexity. The convexity of N (r) and C(p ) enure an efficient numerica oution for the network optimization probem (1). Define the dua objective function g(λ) = L(r, f, p, c, λ). (8) r,f,p,c The imization above can be decompoed into two ubprobem (5) (6). The oution of the two ub-probem aow g(λ) to be evauated. Now, by trong duaity, the overa network optimization probem (1) i oved by the foowing dua minimization probem: min g(λ) (9) λ It remain to how that the update tep (7) ove the dua minimization probem. Thi i due to the fact that the update tep are ubgradient update for λ. It i not difficut to how that (c i f i) are ubgradient for λ. Thu, a ong a tep ize ν (t) are choen to be ufficienty ma, the ubgradient update eventuay converge, and it converge to the goba optimum of the overa network optimization probem. Note that the prima-dua agorithm can be impemented in a ditributed fahion, if the ub-probem have ditributed oution. Thi i true becaue in (7), the update of dua variabe λ in the th ink ony require the oca capacity c and the rate of oca fow i f i. IV. SUB-PROBLEM MODULES We have o far propoed a prima-dua oution framework to ove the probem of achieving optima throughput in mutihop wiree network uing joint optimization acro the network and the phyica ayer. It remain to how how routing ub-probem at the network ayer and the power aocation ub-program at the phyica ayer are effectivey oved. We invetigate different aternative oution in different network cenario. At the network ayer, we examine oution with or without the aumption of network coding. At the phyica ayer, we dicu different agorithmic perpective, incuding dua optimization, geometric programming, and cooperative efih game. Thee aternative conit of modue to be readiy pugged into the genera framework that we have propoed. A. Network Layer Modue 1) Routing baed on Muticommodity Fow: When network coding i not conidered and data eion are unicat eion, the network ayer i naturay modeed a a muticommodity fow probem. The firt group of contraint in our genera framework, which characterize the dependence of overa throughput on ink fow rate, i then a tandard muticommodity fow poytope. The correponding data routing ub-probem at the network ayer can be oved uing avaiabe oution technique for muticommodity fow probem. ) Routing baed on Tree Packing: When network coding i not conidered and data eion are muticat or broadcat eion, routing i achieved by data forwarding and repication at each wiree node. With thee aumption, each atomic data fow propagate aong a tree that pan every node in the data eion. The imum achievabe throughput can be computed by finding the imum number of pairwie capacity-dijoint tree, in each of which the muticat group remain connected. Such optimization ha a traightforward inear programming formuation with an exponentia number of tree capacity variabe. In the cae of broadcat eion, uch a probem correpond to the panning tree packing probem, in which we can work on the dua and empoy minimum panning tree agorithm a the eparation orace. In the cae of muticat eion, the probem correpond to the Steiner tree packing probem, where thi approach doe not work effectivey. Thi i due to the fact that we need to ove the minimum Steiner tree probem in the dua, which i exacty a hard a the Steiner tree packing probem itef [15].
3) Muticat Routing with Network Coding: With the advantage of network coding [], we can mode the routing probem at the network ayer a foow: r,f,e U(r).t. r i O(n) i e i,j I(Tj i) λ f i (10), i, j, T i j V e i,j f i, i, j, E e i,j = e i,j, i, j, n V \{ i, T i j } I(n) f i 0, e i,j 0, r i 0 The firt inequaity repreent the contraint that the ith eion muticat rate r i i e than or equa to the um of a the conceptua fow rate from ource i to each of it jth detination Tj i. Here, ei,j i the conceptua fow rate on ink in the ith muticat eion to it jth detination Tj i. The econd inequaity mean that the actua fow rate f i of eion i on ink i the imum of a the conceptua fow from ource to detination in that eion. It i the advantage of network coding that aow u to conider the imum, rather than the ummation, of the conceptua fow rate. The third equaity contraint repreent the aw of fow conervation for conceptua fow, where I(n) i defined a the et of ink that are incoming to node n; and O(n) i the et of ink that are outgoing from node n. Theorem : For a data network with mutipe muticat eion, the imum utiity and it correponding optima routing trategy can be computed efficienty and in a ditributed fahion. Proof: Firt, the utiity function i a concave function. Second, it i eay to prove that the network coding region contraint (inear contraint) i a convex et. Therefore, oving the ub-probem (10) i a convex optimization probem, which can be oved efficienty. A ditributed oution i ao poibe by further reaxation. For more detaied dicuion, refer to the conceptua fow approach preented in [1] [5]. B. Phyica Layer Modue Interference management i one of the main chaenge in phyica ayer deign of wiree network. A key concept in phyica ayer i the capacity region (rigorouy peaking, the achievabe rate region), which characterize a tradeoff between achievabe rate at different ink. The capacity region optimization probem in the phyica ayer may be formuated a foow p λ c (11).t. c = og (1 + SINR ) G SINR = p j G j p j + σ p p n, n O(n) where c i the capacity of ink, SINR i the igna to interference and noie ratio of ink, G, p, and σ are the ink gain, power, and noie, repectivey. G j i the interference gain from ink j to ink. Each node ha a power budget p n,. Becaue of the interference, the power aocation probem (11) i a non-convex optimization probem which i inherenty difficut to ove. In thi ubection, we dicu three recent technique to eae the way of characterizing the feaibe capacity region: the dua optimization method, geometric programming, and the game theoretic approach. 1) Dua Optimization Method: The main idea of the dua optimization method [7] i to recognize that athough the contraint of (11) are not convex by itef, the time-haring verion of thee contraint away i. Further, the time-haring verion of the probem can ometime be more efficienty oved in the dua domain by oving the Lagrangian dua probem g(µ) = λ c + µ n p n, p, p n O(n) for each fixed µ and by adjuting µ via a ubgradient update. Optimizing for g(µ), athough ti not trivia, i often eaier than oving the origina probem, a g(µ) i away convex. (The exact evauation of g(µ) may ti take exponentia compexity. However, efficient and ub-optima agorithm can often be ued to evauate g(µ) approximatey.) Under the time (frequency) haring condition, the minima vaue of g(µ) over a poitive µ i equa to the optima oution of (11) min µ g(µ) = p λ c A in practica phyica-ayer ytem deign, time-haring can often be impemented either directy or via frequency haring uing (for exampe with OFDM moduation), the above method give an efficient way to ove the capacity region imization probem (11). ) Geometric Programming: Recent deveopment in geometric programming how that, in high SINR cenario, oving the probem (11) can be efficienty accompihed by convex programming [8]. The idea i to firt approximate the ink capacity rate c = og(1+sinr ) og(sinr ) if the SINR i much arger than 1. Then by ogarithmic tranformation of power vector p = og(p ), the tranformed probem (1) i a convex function over variabe p: p : λ c (1) ( ).t. : c = og SINR SINR G e p = j G e p j j + σ O(n) e p pn,,, n
Furthermore, Chiang [8] ha propoed a ditributed power aocation agorithm with gradient tep ize κ a foow: p (t+1) = p (t) + κ λ p (t) λ G G p (t) SINR (t) 3) The Game Theoretic Approach: In thi ection, we expore way to approximate the oution of the non-convex achievabe rate imization probem by game theory. In a power contro game, each ink i modeed a a payer with an aim of imizing it utiity function. The main idea here i to deign a et of utiity function o that the competitive equiibrium of the game i approximatey the goba optimum. A reaching the competitive equiibrium of a game i typicay computationay efficient and amenabe to ditributed impementation, thi give u an effective mean of approximatey oving the phyica ayer power contro probem. Thi type of game i inherenty different from traditiona efih game in that by pretending to be efih, node actuay hep achieve the joint ocia wefare. Such a game i referred to a a cooperative efih game. In conventiona game theoretica approache for power contro [16], each ink ue it own achievabe rate a it utiity function (whie treating a other uer a interference). Competitive equiibria in uch a game may not correpond to deirabe operating point, epeciay when the interference eve i high. Thi i typified by the we-known prioner diemma. The main idea of thi ection i to modify the utiity function of each ink to incude not ony it own achievabe rate but ao the detrimenta effect of the interference each ink caue on other ink. Under thee modified fictitiou utiity function, each ink then ha an incentive to ette on a power eve that trike a baance between imizing it own rate and minimizing the interference caued to other. In a ditributed impementation, the amount of interference caued by each ink ha to be etimated by it neighbor. Thi motivate u to propoe a meage-paing mechanim with which the interference information can be communicated between the ink via a ide channe. Mathematicay, we propoe the foowing utiity function for each ink : ( ) G p Q = λ og 1 + j k G jp j + σ m p µ n p (13) where m i the dua variabe ummarizing the effect of interference from a other ink and µ n i the dua variabe that indicate the price of tranmitter power at node n. A enibe choice for m i the derivative m = c. (14) p In other word, m i the rate at which other uer achievabe data rate decreae with an additiona amount of power. The power price µ n refect how tight the reource at node n i being utiized by it outgoing ink under the contraint O(n) p p n,. The foowing agorithm impement thi game theoretica power contro cheme where each ink payer efihy imize Q, whie continuouy updating the meage. Agorithm : Meage Paing Game Agorithm 1) Initiaize p (0), m (0), µ (0). Set t = 0. ) Set p (τ0) = p (t). Set i = 0, iterativey update p (τi) a foow: p (τi+1) = m (t) λ + µ (t) n j G j p (τi) j σ G G Repeat unti p (τi) converge. Set p (t+1) = p (τi). 3) Update price µ n via ubgradient µ (t+1) n = µ (t) n + γ n (t) p n, O(n) p (t) 4) Update the meage m uing (14) m (t+1) = SINR (t+1) G λ G p (t+1) 5) Repeat (-4) unti convergence. SINR (t+1) 1 + SINR (t+1) The power update in tep () i baed on the foowing. At each tep, each payer trie to imize it own utiity Q whie auming the power eve for a other payer and the meage are fixed. The expreion for optima p i obtained by etting the derivative Q with repect to p to zero. Such a ocay optima p trike a baance between imizing it own rate and minimizing it effect for other uer (which i taken into account via m ). For exampe, a arge vaue for m indicate that ink i producing evere interference to other ink. Thi i refected in the power update a a arge m ead to a ower p. Simiary, the vaue of the pricing variabe µ n indicate the tightne of the per-node power contraint. A high vaue for µ n igna that the uppy for power i tight and it entice ink to reduce it power. Finay, the demand for ink capacity from the upper ayer i refected in the network ayer hadow price λ. A arge vaue for λ indicate that a higher capacity rate in the th ink i needed to upport upper ayer traffic, and it prompt the phyica ayer to increae it power. Athough each payer appear to be efih in imizing it own utiity ony, becaue the utiity function incorporate ocia wefare, the Nah equiibrium of thi game i in fact a cooperative ocia optimum. The meage-paing agorithm can be impemented in a ditributed fahion. Thi i becaue the meage can be ocay coected and broadcat to the neighbor of each ink. Athough thi power contro game doe not neceariy converge to the goba optimum, experimenta evidence ugget that it perform very we in practice. Finay, we have the foowing reut on the condition for convergence. Theorem 3: Agorithm away converge, and it converge to a tabe Nah equiibrium of the meage paing game, if the aboute vaue of eigenvaue of dynamic tabiity matrix are e than one. Due to pace contraint, a detaied anayi on the exitence, uniquene, tabiity and efficiency o for the Nah equiibrium in thi meage paing game i omitted.
S 14 Muticat Rate 3 Muticat Rate 1 7 1 3 6 4 5 8 9 10 11 Rate 10 8 6 4 0 0 0 40 60 80 100 (a) Low Interference Rate.5 1.5 0 100 00 300 400 (b) High Interference 1 13 14 15 Fig.. Convergence of the muticat rate. Fig. 1. 16 T1 The network topoogy. T3 17 T 18 Price 0.1 0.08 0.06 0.04 0.0 Shadow Price Price 0.1 0.09 0.08 0.07 0.06 0.05 0.04 Shadow Price V. ILLUSTRATIVE EXAMPLES We iutrate an exampe of a inge eion muticat in Fig. 1. The ource S attempt to etabih a eion with imum muticat throughput to the three ink T 1, T, T 3, with network coding. The mode for the phyica ayer i an interference channe. An OFDM tranmiion cheme i imuated. The channe gain and interference are randomy generated. We define interference degree (ID) a the average ratio between the deired ink gain and the um of a interference gain. Both the ow interference cae (ID = 1dB) and the high interference cae (ID = db) are invetigated. We ue the propoed prima-dua agorithm to achieve the optima oution for the joint routing and power aocation probem (1). Specificay, a ditributed meage paing game agorithm i empoyed in order to find the achievabe rate region. Fig. iutrate the muticat rate imization proce. In both high and ow interference cae, the muticat rate continuouy converge to the optima oution. However, the convergence peed i different. Convergence i much fater in the ow interference cenario (i.e., 60 iteration) than in high interference cenario (i.e., 00 iteration). Thi i becaue that in the ow interference cae, it i not very important for the ink to exchange meage in order to arrive at a good tradeoff. Intead, each ink impy imize it own ink capacity to upport the network traffic. The convergence proce for the cro-ayer dua variabe i iutrated in Fig. 3. The dua variabe (hadow price) contro the inter-ayer interface o that both routing in the network ayer and power aocation in the phyica ayer can reach an optima matching point. A the hadow price converge, the entire ytem reache an optima oution. Fig. 4 how the matching proce between the network ayer fow and the phyica ayer rate. At the beginning, the network fow ociate in order to find a good routing trategy for each et of phyica-ayer rate. At the ame time, phyica ayer rate increae and decreae among themeve in order Rate 0 0 0 40 60 80 100 Fig. 3. 1 10 8 6 4 (a) Low Interference Convergence of cro-ayer dua variabe. Capacity Support v. Network Fow Capacity Support Network Fow 0 0 0 40 60 80 100 Fig. 4. (a) Low Interference 0.03 0 100 00 300 400 Rate.5 1.5 1 0.5 (b) High Interference Capacity Support v. Network Fow Capacity Support Network Fow 0 0 100 00 300 400 (b) High Interference Convergence between network fow rate and capacity upport. to upport network ayer traffic. Thi proce i coordinated by hadow price a hown in Fig. 3. Eventuay, the network fow and capacity rate agree. Thi oution i optima, in the ene that the phyica ayer come up with the bet reource aocation whie the network ayer route the bet path from the ource to mutipe ink. Together, the muticat rate utiity function i imized. For exampe, for the network in Fig. 1, the fina oution in the high interference cae, network fow in each ink are f 1 = f = 1.98, f 3 =... = f 18 = 0.99; the capacity rate are c 1 = c = 1.98, c 3 =... = c 18 = 0.99; and the muticat rate i r =.97. Finay, it i intereting to point out that our agorithm i energy efficient becaue there i no wated capacity rate in the ytem. A capacity rate exacty upport the network fow (f = c ). Conequenty, a the hadow price are non-zero, which mean the capacity contraint are a active in the origina probem (1). Thi oution ha a -fow min-cut interpretation. If we normaize the throughput, the optima fow and capacity rate wi a be one unit except for ink 1 and, where it i
Fig. 5. a+c a a+c a, c T1 a+c i a a a+b (a, b, c) S T3 a+b a+b b a+b c b, c T Tranmiion cheme with network coding. two unit. A we can ee from the optimization oution, the ource can end three unit of information in tota to each ink, where the -fow rate i exacty equa to the min-cut bound a hown in Fig. 5. We further how a network coding cheme to achieve thi. In our exampe, ource S ha three unit (a, b, c) to end, and each of the ink (T 1, T, T 3 ) can exacty receive them, by uing the fow oution and coding cheme a iutrated in Fig. 5. VI. LIMITATIONS AND EXTENSIONS Thi paper propoe a genera modeing and oution framework for the throughput optimization probem for mutihop wiree network. In thi ection, we point out evera imitation and poibe extenion of the current framework. Inter-eion network coding for mutipe data eion i not conidered in our framework. However, intereion coding provide ony margina throughput gain [1], whie it render the data routing ub-probem NPhard. Thu, ignoring uch poibiitie eem jutifiabe. One of the main phyica ayer aumption in thi paper i that each ink tranmit and receive igna independenty. Poibiitie of muti-acce, broadcat or reay communication are not conidered. The mode preented in thi paper i reaitic in an ad-hoc network where no ynchronization between the node i poibe and interference i away regarded a noie. However, when a moderate amount of node cooperation can be impemented (e.g., a in [17]), the utiization of mutiuer technique in the phyica ayer i expected to provide further gain. b b c c network coding, data routing, and wiree interference can be jointy conidered to achieve the overa optima performance. Our oution framework decompoe the optimization probem into maer ub-probem: data routing at the network ayer and power aocation at the phyica ayer. Modeing and oution agorithm for each ub-probem can be eaiy tuned according to a pecific networking technoogy, a we a avaiabe optimization technique. REFERENCES [1] Z. Li, B. Li, D. Jiang, and L. C. Lau, On Achieving Optima Throughput with Network Coding, Proc. of IEEE INFOCOM, 005. [] R. Ahwede, N. Cai, S. R. Li, and R. W. Yeung, Network Information Fow, IEEE Tran. on Information Theory, vo. 46, no. 4, pp. 104 116, Juy 000. [3] R. Koetter and M. Medard, An Agebraic Approach to Network Coding, IEEE Tran. on Networking, vo. 11, no. 5, pp. 78 795, October 003. [4] S. R. Li, R. W. Yeung, and N. Cai, Linear Network Coding, IEEE Tran. on Information Theory, vo. 49, 003. [5] Z. Li and B. Li, Efficient and Ditributed Computation of Maximum Muticat Rate, Proc. of IEEE INFOCOM, 005. [6] H. Luo, S. Lu, and V. Bharghavan, A New Mode for Packet Scheduing in Mutihop Wiree Network, in Proceeding of the 6th Annua ACM Internationa Conference on Mobie Computing and Networking (MobiCom), 000. [7] W. Yu, R. Liu, and R. Cendrion, Dua Optimization Method for Mutiuer Orthogona Frequency Diviion Mutipex Sytem, IEEE Gobecom, 004. [8] M. Chiang, Baancing Tranport and Phyica Layer in Wiree Mutihop Network: Jointy Optima Congetion Contro and Power Contro, IEEE J. Se. Area Comm, vo. 3, no. 1, January 005. [9] S. H. Low, A Duaity Mode of TCP and Queue Management Agorithm, IEEE/ACM Tran. on Networking, vo. 11, no. 4, pp. 55 56, Augut 003. [10] J. Wang, L. Li, S. H. Low, and J. C. Doye, Cro-ayer Optimization in TCP/IP Network, IEEE/ACM Tran. on Networking, 005. [11] L. Xiao, M. Johanon, and S. Boyd, Simutaneou Routing and Reource Aocation for Wiree Network, Proc. of the Fourth Aian Contro Conference, 00. [1] M. Johanon and L. Xiao, Cro-ayer Optimization of Wiree Network Uing Noninear Coumn Generation, Technica report IR- S3-REG-030, Roya Intitute of Technoogy, Sweden, 004. [13] W. Yu and J. Yuan, Joint Source Coding, Routing and Reource Aocation for Wiree Senor Network, IEEE Internationa Conference on Communication (ICC), May 005. [14] Y. Wu, P. A. Chou, Q. Zhang, K. Jain, W. Zhu, and S. Y. Kung, Network Panning in Wiree Ad Hoc Network: A Cro-Layer Approach, IEEE J. Se. Area Comm, vo. 3, no. 1, January 005. [15] K. Jain, M. Mahdian, and M. R. Saavatipour, Packing Steiner Tree, in Proceeding of the 10th Annua ACM-SIAM Sympoium on Dicrete Agorithm (SODA), 003. [16] W. Yu, G. Gini, and J. Cioffi, Ditributed Mutiuer Power Contro for Digita Subcriber Line, IEEE J. Se. Area Comm, vo. 0, no. 5, pp. 1105 1115, 00. [17] J. E. Wieethier, G. D. Nguyen, and A. Ephremide, On the Contruction of Energy-Efficient Broadcat and Muticat Tree in Wiree Network, Proc. IEEE INFOCOM, 000. VII. CONCLUSIONS In thi paper, we have propoed a genera framework for both modeing and oving the throughput optimization probem in muti-hop wiree network. In our framework,