Efficient load-balancing routing for wireless mesh networks

Transcription

1 Computer Networks 51 (007) Efficient lod-blncing routing for wireless mesh networks Yigl Bejerno, Seung-Je Hn b, *,1, Amit Kumr c Bell Lbortories, Lucent Technologies, NJ, United Sttes b Deprtment of Computer Science, Yonsei University, Seoul, Republic of Kore c Deprtment of Computer Science nd Engineering, Indin Institute of Technology Delhi, New Delhi, Indi Received 1 April 006; ccepted 5 September 006 Avilble online 0 November 006 Responsible Editor: E. Knightly Abstrct Wireless mesh networks (WMNs) consist of sttic wireless routers, some of which, clled gtewys, re directly connected to the wired infrstructure. User sttions re connected to the wired infrstructure vi wireless routers. This pper presents simple nd effective mngement rchitecture for WMNs, termed configurble ccess network (CAN). Under this rchitecture, the control function is seprted from the switching function, so tht the former is performed by n network opertion center (NOC) which is locted in the wired infrstructure. The NOC monitors the network topology nd user performnce requirements, from which it computes pth between ech wireless router nd gtewy, nd lloctes fir bndwidth for crrying the ssocited trffic long the selected route. By performing such functions in the NOC, we offlod the network mngement overhed from wireless routers, nd enble the deployment of simple/low-cost wireless routers. Our gol is to mximize the network utiliztion by blncing the trffic lod, while providing fir service nd qulity of service (QoS) gurntees to the users. Since, this problem is NP-hrd, we devise pproximtion lgorithms tht provide gurntees on the qulity of the pproximted solutions ginst the optiml solutions. The simultions show tht the results of our lgorithms re very close to the optiml solutions. Ó 006 Published by Elsevier B.V. Keywords: Wireless mesh network; Lod-blncing routing; Firness; Approximtion lgorithms 1. Introduction The recent dvnce of wireless communiction technologies hs prompted flourish of new kind E-mil ddress: sjhn@cs.yonsei.c.kr (S.-J. Hn). 1 This work is in prt supported by KOSEF Grnt R of multi-hop wireless network rchitecture, clled wireless mesh networks (WMNs). WMNs typiclly comprise number of sttic wireless routers tht re ttched to relible sources of energy. The wireless routers re interconnected with ech other vi wireless links nd provide communiction services to mobile or sttic users in their vicinities. Some of the routers re directly connected to fixed infrstructure (i.e., wired network like the Internet) nd serve s gtewys for other wireless routers /$ - see front mtter Ó 006 Published by Elsevier B.V. doi: /j.comnet

2 Y. Bejerno et l. / Computer Networks 51 (007) Commercil wireless mesh solutions re currently offered by such vendors s BelAir Networks ( Mesh Dynmics ( nd Air Mtrix (www. irmtrix.net), minly for pplictions like wireless brodbnd ccess networks nd disster recovery networks. For effectively serving such pplictions, the network utiliztion must be mximized while providing firness nd bndwidth gurntees to the users [1]. One of the min concerns for WMNs is the reduction of the overll network cpcity due to interferences between djcent nodes []. To mitigte wireless interferences, severl techniques cn be used including multiple rdios [,], directionl ntenns [] nd MIMO (multiple input multiple output). However, such physicl lyer solutions lone re not enough. To mximize the network utiliztion while preserving firness requirements, efficient routing scheme is criticl [1]. To ddress this need, we propose simple nd effective lod-blnced routing scheme, vi which the network utiliztion is mximized while providing firness nd bndwidth gurntees. The proposed scheme is comptible with ny physicl lyer solution for interference mitigtion Relted work Mngement of wireless multi-hop networks hs been n ctive reserch re nd numerous routing lgorithms hve been proposed. Comprehensive surveys on WMNs nd routing in multi-hop wireless networks cn be found in [5,6]. Most of the routing schemes for multi-hop wireless networks im t such environments s bttlefield d-hoc networks, nd the typicl objective is to mintin the communiction links between mobile sttions. Providing connectivity, however, is not sufficient for WMNs in which the users demnd Qulity of Service (QoS) gurntees comprble to the wired networks. Severl routing schemes hve been proposed for WMNs. Here, we mention only those studies tht re directly relevnt to our work. In [7], De couto et l. introduce the expected trnsmission count (ETX) metric tht enbles existing routing lgorithms to find high performnce pths between source destintion pirs where single rdio chnnel is used. Drves et l. propose in [] new pth metric, clled weighted cumultive expected trnsmission time (WCETT), tht explicitly ccounts for the interference mong links using the sme chnnel. Then, they incorporte the WCETT metric into source-route link-stte-like routing tht exploits the dvntge of the multiple rdios. In [8,10] the uthors ddress both the chnnel ssignment nd the routing problems of WMNs with multiple rdios. They present chnnel ssignment heuristics tht mintin connectivity requirements while minimizing the interferences. In [8], Tng et l. present bndwidth wre routing (BAR) lgorithm tht selects either single pth or multiple pths for ech incoming session request, which mximizes bndwidth lloction of tht session. In [10], Kysnur nd Vidy present n enhnced shortest pth routing method. Beside the hop-count of the pth, it tkes into ccount dditionl spects such s the interferences from other nodes tht use the sme chnnel. In summry, the bovementioned routing schemes increse the bndwidth lloction of individul session requests nd typiclly chieve high overll network utiliztion. However, they do not consider the issue of firness in the bndwidth lloction, so tht wireless sttions tht re severl hops wy from the gtewy my suffer from low bndwidth. For the fir prtition of the network resources, severl studies use blnced trees rooted t the gtewys nd route the trffic long the tree pths. These studies commonly ssume WMNs in which multiple rdios re used while using different interference models. In [11], Hsio et l. consider grid mesh topology with single gtewy where ech node hs interference-free point-to-point connection to its immedite neighbors. Under this setting, centrlized heuristic is proposed for clculting lod-blnced H-trees tht llocte the sme bndwidth to ll the nodes. In [1,], both chnnel ssignment nd routing re considered together. In [1], He et l. present heuristic for clculting lod-blnced shortest pth tree by tking the trffic flows into ccount. After clculting the tree, they perform chnnel ssignment to efficiently utilize the wireless links. In [], Rniwl nd Chiueh present distributed lgorithms tht use only locl trffic lod informtion for determining the chnnel ssignment nd the tree topology. Though these existing heuristics improve the firness of the llocted bndwidth to the nodes, they do not provide ny gurntee on the qulity of the solutions ginst the optimlity. Some studies exploit the dvntge of multiplepth routing. In [1], Jin et l. consider the

3 5 Y. Bejerno et l. / Computer Networks 51 (007) problem of optiml multi-pth routing, where the interferences re modeled by conflict grph. A similr problem is ddressed by Kodilm nd Nndgopl in [1]. This study dels with the joint problem of routing nd scheduling of multi-pth flows, ssuming tht ech wireless sttion is equipped with single rdio but the sttions use orthogonl chnnels in order to void interferences. In [15], the uthors extend their result for the cse of multiple rdios. These studies hve shown tht multi-pth routing mximizes overll trffic flow while providing fir service nd bndwidth gurntees. However, these methods fce difficulties in the trffic mngement, since the trffic between ech source destintion pir my be divided into multiple smll flows nd they generte high communiction nd computtion overhed on the network nodes [16,17]. In [17], Gnjli nd Keshvrzin clim tht in prctice the lod distribution obtined by multi-pth routing is essentilly similr to the single pth routing, unless very lrge number of pths re used (which is prcticlly infesible). 1.. Our contributions In this pper, we present simple nd effective mngement rchitecture for WMNs, termed configurble ccess network (CAN). This rchitecture is inspired by the observtion tht WMNs usully serve s ccess networks [1,] nd consequently ll nodes of WMNs re ccessible from the wired infrstructure. Such WMNs cn be mnged by externl sttions, termed network opertion center (NOC). The use of externl NOCs offlods the mngement overhed from the wireless routers nd thus reduces their complexity. This centrlized mngement pproch tht seprtes the switching functions from the control functions hve lredy been dopted in wired networks, e.g., the softswitch model of converged networks [18] nd the softrouter rchitecture for IP networks [19]. In prctice, some rudimentry centrlized mngement schemes with externl NOCs hve lredy been dopted by some WMN commercil products such s BelAir Networks nd Air Mtrix. We present lgorithms for single-pth routing nd bndwidth lloction tht cn be incorported into the CAN rchitecture. Our lgorithms chieve ner-optiml fir bndwidth lloction without the drwbcks of the multi-pth routing. The singlepth pproch hs severl dvntges over the multi-pth pproch: (i) it simplifies the trffic control, (ii) it mintins the pcket delivery order, (iii) it enbles the deployment of efficient compression schemes. Compression schemes, such s RObust Heder Compression (ROHC) scheme [0], yield significnt sving in bndwidth consumption by incresing network throughput without requiring the ctul increse of the network cpcity. However, high compression rtio cn be obtined only when the pckets mintin their order nd trverse through the sme gtewy, which is not lwys possible in cse of multi-pth flows. Vi extensive simultions, we exmine the performnce of the singe-pth pproch ginst the multi-pth optiml solutions, without considering the gin by compression. The simultions show tht the bndwidth lloction obtined by our single-pth pproch is very close to the multi-pth optiml solution, in prticulr when the number of nodes nd users is high. This confirms the observtion of Gnjli nd Keshvrzin in [17]. Our simultions show tht the selected pths re typiclly long the shortest pths between the nodes nd the gtewys, resulting miniml end-to-end dely which is criticl for reltime pplictions. Our lgorithms find set of pths tht mximize the normlized bndwidth lloction of the nodes, which is defined s the miniml bndwidth llocted to ech user. This objective is ligned with the firness reference model described in [1], tht ims t the fir bndwidth lloction to the users independent of their loctions or their distnces from gtewy. Since this problem is NP-hrd, we develop severl lgorithms tht provide different pproximtion rtios under different settings. All these lgorithms initilly clculte the optiml multi-pth flow solutions. Then, we utilize the single-source unsplittble flow lgorithm by Dinitz et l. [] to extrct the single-pth routing from the multi-pth solutions. Ech wireless router (termed node) is ssocited with weight d v, tht represents the number of users who re ssocited with the node. We first present n lgorithm tht finds n optiml solution when ll nodes hve the sme weights nd ll wireless links hve the sme cpcity. When the nodes hve different weights, this lgorithm gurntees t lest hlf of the optiml normlized bndwidth. Then, we del with the cse tht links cn hve rbitrry cpcities nd the node weights re bounded by d min nd d mx. In this cse, we chieve 1 + -pproximtion fctor, where ¼ dmx d min, by forwrding the trffic only long links with

4 Y. Bejerno et l. / Computer Networks 51 (007) sufficient cpcity. This is ctully -pproximtion for the fir ggregted bndwidth lloction objective presented in [1]. For the cse of rbitrry node weights nd link cpcities, we provide 5-pproximtion lgorithm. To the best of our knowledge, our scheme is the first work tht provides gurntees on the qulity of single-pth routing solutions in the context of wireless multi-hop routing. Our gol is mximizing the network utiliztion, while providing firness. Since, wireless chnnel qulity my frequently chnge due to interferences nd fst fding effects [9], it is dunting tsk to chieve this gol in short-term scle, i.e., to optimize the network performnce t ny given moment. Frequent chnges of the trffic routes my led to route oscilltions which cn severely degrde the overll performnce. We focus on the long-term firness by occsionlly modifying the trffic routes to ddress topology chnges or user mobility.. Network model This pper considers WMNs which comprise of sttic or qusi-sttic wireless mesh routers, termed nodes, nd mobile or sttic user sttions, termed users. Some nodes, referred s gtewys, re equipped with bckhul links to fixed infrstructure network nd serve s ccess gtewys for other nodes. Ech node is typiclly equipped with one or more omni-directionl ntenns (e.g., using IEEE b or g) to provide the network connectivity to the users in its vicinity. It lso hs dditionl bckhul rdio interfces for point-to-point connectivity with its djcent nodes (e.g., using IEEE or IEEE 80.16). The bckhul rdio interfces my use directionl ntenns. The network is represented by n undirected grph G(V [ {},E), where the grph nodes, V, represent the wireless routers nd is virtul node tht corresponds to the fixed infrstructure. We denote by E the network links including the bckhul links between the gtewys nd the infrstructure s well s the wireless links between nodes. Ech link e E hs cpcity (bit-rte) C e for both direction. Since, chnnel qulity my be time-vrying, we ssume the C e is the verge link cpcity or its lower bound, depending on the QoS requirements. For instnce, for dely-tolernt pplictions C e my use the verge chnnel qulity, while for rel-time pplictions the lower bound my be used. By ssuming directionl ntenns nd sufficient number of wireless chnnels, we ignore the interference between chnnels. Ech node v V is ssocited with weight d v Z þ tht is proportionl to its bndwidth requirement. Typiclly, d v indictes the number of users ssocited with node v. Let b v be the bndwidth lloction for node v. b v /d v is the verge bndwidth llocted to ech one of its ssocited users nd referred s the normlized bndwidth, B to the miniml bndwidth tht user my experience, i.e., B ¼ min vv b v =d v. We denote by d mx = mx vu d v nd d min = min vu d v the mximl nd miniml weights of the nodes, respectively. The neighborhood of node v V, denoted by N(v), is the set of nodes with whom v hs connections. When node v is gtewy. the virtul node is included in the neighborhood of v. In this pper, we ssume tht the user trffic lwys flows from or to the fixed infrstructure, so tht the trffic lwys trverses through the gtewys, while non-gtewy nodes serve s relys.. The CAN rchitecture One of the min chllenges in WMNs is chieving high network utiliztion nd long-term firness simultneously. This objective is crucil s nodes tht re severl hops wy from their serving gtewys my experience low service qulity or even been strved. It requires efficient routing nd bndwidth lloction. The lgorithms for route selection nd bndwidth lloction will be treted in Section, nd this section describes the CAN rchitecture which cn fcilitte such lgorithms..1. Overview The CAN rchitecture is inspired by the following observtion: WMNs re minly used s ccess networks for sending/receiving informtion to/from the users vi wireless routers. As ll nodes re lwys connected to wired infrstructure, it is possible to shift the resource mngement tsks, tht my be done by the nodes, to NOC, which is locted in the wired infrstructure. In prticulr, we propose tht the NOC determine the routes between the nodes nd the gtewy s well s llocting pproprite bndwidth for ech trffic flow. For instnce, eight (or more) non-overlpping chnnels re vilble for IEEE

5 5 Y. Bejerno et l. / Computer Networks 51 (007) To this end, the NOC needs to know the network topology, the link cpcities nd the nodes weights. Bsed on the collected informtion, it determines routing nd bndwidth lloction, nd configures the nodes ccordingly. In WMNs the wireless nodes re typiclly sttic (or qusi-sttic) nd connected to relible sources of energy. Such networks experience reltively infrequently topology chnges tht cn esily be reported to the NOC. The CAN rchitecture hs severl dvntges over the existing methods for WMNs tht re typiclly bsed on distributed selforgnized solutions. In contrst to the existing solutions tht mostly focus on mintining connectivity, our solution mximizes the fir bndwidth lloction to the users. Moreover, since the routing nd bndwidth lloction decisions re mde t the NOC, the wireless routers need only modest computtion cpbilities. The CAN rchitecture lso reduces the communiction overhed, since topology chnges re informed only to the NOC insted of brodcst to ll other nodes. These benefits mke the CAN rchitecture prticulrly ttrctive for lrge-scle WMNs... Inferring the network topology Consider wireless mesh network modeled by grph G(V [ {},E) s described in Section. All nodes re ssumed to know their immedite neighbors nd the NOC is connected to ll the gtewys. Upon the ctivtion, NOC strts to infer the network topology, the link cpcities, nd the node weights by sending out queries. At first, by querying the gtewys, the NOC obtins the ddresses of ll the nodes tht re one hop wy from the gtewys. Similrly, by querying ll the gtewys neighbors, the NOC lerns bout ll the nodes tht re two hops wy from the gtewys. Thus, by performing bredth first serch (BFS) the NOC discovers the network topology lyer by lyer. During this topology discovery process, the NOC communictes with node by using source-routing method, i.e., ech control messge crries its complete pth to its destintion nd the reply is returned on the reversed pth... Single vs. multiple trffic pths One of the min decisions tht the NOC mkes is computing the trffic routes for ech individul user. Either single pth or multiple pths cn be used for ech user trffic flow. Generlly speking, the multiple-pth pproch mximizes the network utiliztion nd the fir shre of ech user. This pproch is lso more resilience to filures. The price for such benefits is the complexity in the trffic mngement. For instnce, it my require dividing the pckets of single flow to multiple routes, mnging lrge number of smll flows nd mintining the pckets order t the ggregtion point. Therefore, we believe tht routing the trffic of ech flow long single pth is more prgmtic pproch. In single-pth pproch, for ech flow, the NOC selects single pth, termed virtul connection (VC), tht hs dedicted bndwidth lloction nd unique VC identifier. Pckets of this flow crries in their heder the VC identifier, which is used by the intermedite nodes to forwrd the pckets long the selected pth. Such pcket forwrding method is used, for instnce, by ATM [] nd MPLS []. In ddition to the simplicity in the network mngement nd trffic control, the single-pth pproch hs other importnt dvntges over multiple-pth pproch. It mintins the pcket delivery order, which is importnt to the performnce of mny protocols such s TCP. Mintining pcket order is in prticulrly crucil for the use of heder comprison lgorithms. For instnce, in the cse of voice over IP (VoIP) trffic, mintining the pcket order is essentil for efficient utiliztion of the Robust Heder Compression (RoHC) protocol [0], which reduces the pcket size from 6 bytes (8 bytes in the cse of IPv6) to only bytes, ssuming voice pylod of bytes per pcket. Such compression cn significntly increse the network utiliztion. Furthermore, our simultion results in Section 5 show tht in prcticl settings, efficient single-pth solutions yield network utiliztion nd normlized bndwidth similr to the optiml solutions using multiple pth flows. Tking the gin by heder compression into ccount, the single-pth solutions cn outperform the multiple-pth solutions. In this pper we consider two vrints of singlepth solutions. In the first vrint, termed ggregted flow model, the ggregted trffic of ll the users tht re ssocited with given node v is forwrded long single pth. While, the second vrint, termed user flow model, the trffic of ech By using lbel swpping technique, ech VC need to hve unique VC identifier t ech node.

6 Y. Bejerno et l. / Computer Networks 51 (007) user is routed seprtely over its own pth. On one hnd, the user flow model my utilize the network resources more efficiently thnks to its multiple-pth nture. It my be beneficil in prticulr for pplictions of constnt bit-rte user flows like voice clls. On the other hnd, since ech user hs dedicted bndwidth lloction, the user flow model my be less desirble for the pplictions tht hve bursty trffic chrcteristics. For such bursty trffic, the ggregted flow model my be more efficient by utilizing sttisticl multiplexing mong multiple users tht shre the sme VC. The single-pth solution is more vulnerble to the network filures. This problem cn be overcome by setting up one or more bckup VC(s) for ech flow, nd in cse of filure the ffected flows re rerouted long the bckup bths. Since the nodes re connected to relible source of energy, we expect filures rrely occur. In this pper we focus on the routing of the primry VCs. An exmple of n ggregted flow route selection is given in Exmple 1. Exmple 1. An exmple of CAN system with two gtewys b nd c, is depicted in Fig. 1. In this exmple the gtewys re ttched to virtul node tht represents the fixed infrstructure. We ssume tht ll the links hve cpcity of 6 nd the weights (i.e., bndwidth demnds) of the nodes re d b =, d g =,d h = nd d i =. For simplicity, we ssume tht the weights of the other nodes {c,d,e,f} re zero. The figure shows the optiml route selection where both bckhul links (,b) nd (,c) re full utilized nd the normlized bndwidth is B ¼ 1... Network configurtion After computing the set of VCs nd determining the bndwidth lloction of ech VC, the NOC configures the nodes ccordingly s follows: first, it builds forwrding tble for ech node v V. This tble contins record for every VC tht trverses through node v with the required informtion for pcket forwrding, such s the VC lbel, the identifictions of the predecessor nd successor nodes nd the VC bndwidth lloction. Second, it updtes the node forwrding tbles by sending dedicted configurtion messges. Finlly, it sends ctivtion messges to the nodes to strt dt forwrding. Due to the user mobility, the node weights nd their bndwidth demnds my chnge. Such chnges my require modifictions of the VC bndwidth lloctions. Since nodes typiclly support multiple users, the movement of user typiclly incurs only smll weight chnges tht cn be esily ddressed by minor djustments of the VC bndwidth lloctions, without chnging the VC routes. When the network configurtion is significntly devited from the optiml settings, the NOC reclcultes the nodes VCs nd reconfigures the network. To mintin the continuous network opertion during the configurtion setting opertion, dul sets of VCs re used. At ech configurtion setting opertion the NOC modifies one VC set, while the other set is used to forwrd trffic. At the end of the opertion, the NOC send messge to ll nodes to switch to the lterntive set. We do not ddress the online bndwidth lloction chnge ny further, nd focus on the optiml route selection. b d e f g h i A node weight c A Virtul Connection Fig. 1. An exmple of CAN system tht utilizes ggregted flow route selection.. Single-pth routing lgorithms Our single-pth routing lgorithms (for primry pths) mximize the normlized bndwidth of the nodes. The problem formultion is given in Section.1. We consider two route selection settings. The first setting, referred s the ggregted flow model, ggregtes trffic of ll the users tht re ssocited with given node v nd routes the ggregted flow long single pth. For this setting, we present three lgorithms tht provide different gurntees on the qulity of the solution depending on the chrcteristics of the underlying networks. In Section., we present polynomil time lgorithm tht finds n optiml route selection when ll nodes hve the sme weight nd ll links hve the sme

7 56 Y. Bejerno et l. / Computer Networks 51 (007) cpcity. This lgorithm lso ensures solution with t lest hlf the normlized bndwidth of the optiml integrl solution, when ll the links hve the sme cpcity but nodes my hve rbitrry weights. In Section. we introduce the concept of rely group nd we use it to construct n (1 + )-pproximtion for the cse of bounded node weights nd rbitrry link cpcities, where is the rtio between the upper nd lower bounds on the node weights. Then, in Section., we present 5-pproximtion lgorithm for the generl cse. In the second setting, termed n user flow model, users who re ssocited with the sme node re not required to shre the sme pth, nd the trffic of ech individul user is routed seprtely (still single pth for ech user). For this model, we present n lgorithm tht ensures -pproximtion in Section The problem sttement Consider grph G(V [ {}, E) s described in Section. We denote by V the set of nodes (i.e., wireless routers), while is virtul node tht represents the fixed infrstructure. Let E be the set of edges comprising both bckhul links nd wireless links between djcent nodes. Ech link e E is ssocited with cpcity C e nd d v is the weight of node v V. We tckle the problem of route selection nd bndwidth lloction tht mximizes the miniml bndwidth llocted to ech user, termed the normlized bndwidth. More specificlly, let P v be the selected route for flow originted by node v (or by user ssocited with node v) nd let b v be the llocted bndwidth to this flow. Then the normlized bndwidth B is defined s the gurnteed miniml bndwidth to ech user, i.e., B ¼ min vv ^dv>0b v =d v. A route selection is termed fesible if the overll bndwidth lloction of ll the routes tht trverse through ny given link e E does not exceed the link cpcity C e. We formlly define the route selection problem s follows: Definition 1 (The bndwidth mximiztion problem). Given grph G(V [ {}, E) with cpcity C e for every link e E nd weight d v for every node v V. Find fesible set of pths {P v } nd the corresponding bndwidth lloctions {b v } between nd every node v V, tht mximize the normlized bndwidth, i.e., B ¼ mx min vv ^dv >0b v =d v. The selected set of pths {P v } nd the corresponding bndwidth lloctions re termed route selection. Theorem 1. The bndwidth mximiztion route selection problem is NP-hrd in the cse of different node weights. Proof. We prove this theorem by presenting polynomil reduction from the prtition problem [5] to the route selection problem. Consider set Q of m > elements with size s i Z þ for every q i Q, nd let X ¼ P q i Q s i=. The prtition problem P looks for P sub set Q 0 Q such tht q i Q 0s i ¼ q i Q Q 0s i ¼ X. We construct grph G(V [ {}, E) with node v i for every element q i Q, nd two gtewys denoted by u 1 nd u, s depicted in Fig.. The weights of the nodes u 1 nd u re 0, nd the weight of every node v i is q i, i.e., d vi ¼ q i. Moreover, let ssume tht the cpcity of the bckhul chnnels (u 1,) nd (u,) re C u1 ; ¼ C u ; ¼ X. We clim tht there is subset Q 0 Q with P q i Q 0s i ¼ X if nd only if there is fesible route selection with normlized bndwidth of 1. Suppose tht there is such prtition Q 0. Then we construct the routes s follows, for every q i Q 0 the pth P vi between the virtul node nd node v i trverses through node u 1, otherwise the pth trverse through node u. Thus, the ggregted weights of ll the nodes tht re served by ech one of the gtewys u 1 nd u is X/. Thus, the normlized bndwidth llocted to ll nodes is 1. Now, suppose tht there is fesible route selection, such tht the normlized bndwidth is 1. Let Q 0 be the set of elements represented by ll the nodes tht their P routes trverse trough node u 1. So the sum q i Q 0s i ¼ X nd this completes the proof. h.. The bsic scheme We now turn to describe our route selection lgorithms for the ggregted flow model. All the proposed lgorithms consist of two steps. In the first step, the lgorithms clculte frctionl solution, lso termed splittble flow solution, tht is llowed to divide ech trffic flow b v into smll flows nd route them long multiple pths. The frctionl routing problem is formulted s single-source v 1 v u 1 v m u Fig.. The grph for the proof of Theorem 1.

8 Y. Bejerno et l. / Computer Networks 51 (007) multiple-destintions flow problem tht mximizes the normlized bndwidth of the users. Recll tht such frctionl solution lwys exists. Then, we utilize rounding lgorithm for obtining n integrl solution, termed n unsplittble flow solution, with single route for ech node flow. The ltter is bsed on the single-source unsplittble flow lgorithm of Dinitz et l. []. Since the rounding lgorithm my clculte link totl flow tht exceed the link cpcity, fter the rounding step we scle the bndwidth lloctions to stisfy the cpcity constrints. Note tht the frctionl solution is n upper bound for the optiml integrl solution...1. The frctionl solution We strt with scheme tht llows ll the nodes to serve s relys. Consider grph G(V [ {}, E)s described bove nd let b v be the bndwidth lloction of every node v V. We denote by F u,v the flow long the link (u,v) from node u to node v nd let C mx v ¼ mx unðvþ C u;v be the mximl cpcity of ny incoming link of node v. Our gol is mximizing the normlized bndwidth lloction defined by B ¼ min vv ^dv>0b v =d v. Thus, the frctionl routing problem cn be formulted s liner progrm (LP) s follows: mx B subject to 8v V ^ d v > 0 : B 6 b v =d v ; ð1þ 8v V ^ d v ¼ 0 : b v ¼ 0; ðþ X 8v V : F u;v ¼ X F v;u þ b v ; ðþ unðvþ unðvþ 8ðu; vþ E : F u;v þ F v;u 6 C u;v ; ðþ 8v V : b v 6 C mx v ; ð5þ 8ðu; vþ E : F u;v P 0; F v;u P 0: ð6þ In this formultion, Constrints (1) nd () ensure tht B lower bounds the normlized bndwidth lloction of every node v V {} with positive weight, while bndwidth is not llocted to nodes with weight zero. Constrint () is the flow conservtion requirement tht ensures tht the mount of flow withdrwn by node v is exctly its bndwidth lloction b v. Constrint () lso ensures tht the ggregted flow originted by the source node is P vv fg b v. Constrint () gurntees cpcity constrint. Finlly, constrint (5) is n lloction constrint to gurntee n upper bound on node llocted bndwidth, such tht node bndwidth lloction does not exceed the mximl cpcity of its edges. The optiml frctionl solution cn be found by using ny LP solver or mximl flow pproch, s described in [6]. Alterntively, pproximtion methods, like the ones described in [7] cn be used to find ner-optiml solutions. It is esy to see tht Corollry 1. The solution of the frctionl routing problem is n upper bound of the normlized bndwidth lloction nd it cn be clculted in polynomil time. Exmple. Consider the CAN system with two gtewys, b nd c, s described in Exmple 1. In this system ll the links hve the sme cpcity 6 nd the node weights re d b =, d g =, d h = nd d i =. For simplicity, we ssume tht the weights of the other nodes {c,d,e,f} re zero. Fig. () presents the frctionl flow solution for the given grph. Recll tht the flows through the links (,b) nd (,c)ref,b = F,c = 6. Thus, the normlized bndwidth lloction to ll the nodes is B ¼ 1. In other words, b b =,b g =,b h = nd b i =.... The rounding lgorithm After clculting frctionl routing, we round the frctionl flows to obtin n integrl solution. Let G 0 (V,E 0 ) be the directed grph induced by the flows of the frctionl solution. A directed link (u,v) E is included in G 0 only if there is strictly positive flow from u to v. Without loss of generlity, we ssume tht G 0 (V,E 0 ) is cyclic grph, s directed cycle cn be eliminted by flow decomposition. Now, we utilize the single-source unsplittble flow lgorithm of Dinitz et l. on the constructed grph 1 b 6 6 d e f 1 1 g h i c b c d e f g h i A link nd its flow. A node weight / Terminl loction. A link without flow. A terminl pth / selected VC. Fig.. The frctionl solution () nd the preliminry terminl shifting phse (b).

9 58 Y. Bejerno et l. / Computer Networks 51 (007) An edge tht is not included in the forwrd pths. G 0. For completeness, we provide concise description of the lgorithm. Its correctness nd performnce nlysis cn be found in []. The rounding lgorithm ssocites token t v with ech node v V {} tht represents trffic flow of size b v. The lgorithm modifies the network flows by moving the tokens bckwrd, until they rech the source node. As token t v is moved bckwrd long n edge e, the flow F e is reduced by b v nd edges with zero flow re eliminted. So, t ny time the network stisfy the flow conservtion requirement in respect to the current loctions of the tokens. Finlly, the lgorithm selects the movement route of the token t v s the route P v of every node v. Let us denote by t v both the token identifier nd its current loction. In preliminry phse, the lgorithm checks for every token t v whether there is n incoming edge e =(u,t v ) with flow greter thn or equl to node s v llocted bndwidth b v. In such cse, it move t v to u nd decreses the flow of e by b v.ife does not crry ny more flow it is removed from the grph. The lgorithm repets this process s much s possible nd retins only tokens tht do not coincide with the source node. Observe tht the resulting instnce mintins degree property such tht the tokens re locted only t nodes with t lest two incoming edges. An exmple of the preliminry phse is given in Fig. (b) for the frctionl solution presented in Fig. (). The rounding lgorithm proceeds in itertions tht ech one consists of three steps. It first finds n lternting cycle, then it ugments the flow long this cycle nd finlly it shifts tokens ccording to movement rule tht keeps the degree property. The lgorithm constructs n lternting cycle by performing tour on the grph edges. It strts t the source node nd cretes forwrd pth by following outgoing edges s long s possible. Since the grph is cyclic, the forwrd pth must end t node with token t v. The lgorithm proceeds by constructing bckwrd pth strting from t v. Since t v hs t lest two incoming edges, the lgorithm chooses n unselected edge nd follows the incoming edge until reching the first node, sy u, tht hs nother outgoing edge. The lgorithm builds nother forwrd pth by following this outgoing edge of u. The process continues in this mnner until it reches node, sy w, tht hs lredy been visited nd close cycle. Thus, the cycle consists of lternting forwrd nd bckwrd pths, s depicted in Fig. () nd (c). Now, the lgorithm modifies the flow long this cycle by shifting smll mount of flow from the forwrd pths to the bckwrd pths in wy tht mintins the flow conservtion requirement. It clcultes two quntities, f nd b. Where, f is the miniml flow of the edges long the forwrd pths nd b is the miniml difference between the flow long link (u,t v ) nd the bndwidth lloction b v for every token t v tht is locted in one of the cycle nodes (or infinity if there re no tokens in the cycle). Then, the shifted mount of flow is the smller of the two, i.e., min( f, b ). Recll tht if the minimum is chieved for f then fter the ugmenttion there is no flow long one of the forwrd edge nd this 1 b d e f 1 1 g h i b c c d e f 1 1 g h i b c d e f 1 1 g h i b c d e f 1 1 g h i b c d e f 1 1 g h i 7 5 b c d e f g h i The finl integrl solution. Terminl. Bckwrd pth. forwrd pth. Fig.. An exmple of clculting n integrl solution.

10 Y. Bejerno et l. / Computer Networks 51 (007) edge is removed. Otherwise, the minimum is obtined for n edge (u,t v ) on bckwrd pth. After the ugmenttion, the flow long this pth is b v nd the lgorithm repets the preliminry phse. So, the token t v is moved long the edges of the bckwrd pth (nd possibly more edges) nd these edges re removed from the grph. The lgorithm ends when ll the tokens rech the AP nd the corresponding pths re determined for the trffic flows. An execution of the lgorithm is illustrted in Exmple. Exmple. Fig. illustrtes few of the steps mde by the rounding lgorithm, while clculting n integrl solution for the frctionl solution given in Fig. (). Fig. () nd (c) presents two clculted lternting-cycles. While, Fig. (b) nd (d) shows the resulting flow fter shifting some flow from the forwrd pths to the bckwrd pths ( units in Fig. (b) nd 1 unit in Fig. (d)). After performing those two flow shifting opertions the token t g cn be moved to the virtul node, s shown in Fig. (e), nd the finl integrl solution is given in Fig. (f). In this solution, link (,b) serves the flows of both nodes b nd g. Thus the ggregted weight of the flows tht trverse through the link (, b) is 7. However, s we clculted in Exmple, the normlized bndwidth of the optiml frctionl solution is B ¼ 1. Thus, if we retin normlized bndwidth of 1, the integrl solution exceeds the cpcity of the link (,b). We overcome this over subscription problem by scling down the normlized bndwidth, s we describe below. In this cse, the selected normlized bndwidth is B ¼ 6 7 nd the bndwidth lloctions re b b ¼ 7 ; b g ¼ 7 ; b h ¼ 7 nd b b ¼ As result, the ggregted flow tht trverses through the link (,b) is 6 nd it stisfies the link cpcity. Exmple demonstrtes tht the rounding lgorithm my not find the optiml integrl solution. In Theorem we bound the devition from the optiml solution. Theorem (From []). The rounding lgorithm finds unsplittble flow such tht the totl flow through ny edge exceeds its initil flow by less thn the mximl llocted bndwidth, b mx ¼ d mx B. From Theorem follows tht fter selecting route P v for ech node v V, some of the link flows my exceed the link cpcities. We overcome this over subscription problem by scling down ll bndwidth lloctions. Let k be the mximl over subscription rtio, i.e., k = mx ee F e /C e. For stisfying the cpcity constrints we scle down ll the bndwidth lloction by fctor of k, such tht the new bndwidth lloction of every node v V is b i v ¼ b v=k.... Algorithm nlysis We nlyze the qulity of the clculted integrl solution reltive to n optiml one for given grph G(V [ {}, E). In our nlysis, we consider the normlized bndwidth B f, B i nd B of the frctionl, integrl nd optiml solutions, respectively. We mesure the link flows of given solution with units of its normlized bndwidth lloction B, termed normlized bndwidth unit, i.e., B ¼ 1. Thus, the flow F e of link e E represents the number of normlized bndwidth unit tht trverse through this link. Obviously, the normlized bndwidth of solution is mximized when the mximl link flow is minimized. Consequently, we denote by F f e, F i e nd F e the flows tht trverse through link e in the frctionl, integrl nd optiml solutions mesured with the corresponding normlized bndwidth units, B f, B i nd B, respectively. Hereby, we consider instnces where ll links hve the sme cpcity. Theorem. If the weight of every node v V is either zero or constnt d nd ll the links hve the sme cpcity, C, then the bsic lgorithm finds n optiml integrl solution. Proof. Let us consider the optiml frctionl solution nd let link e 0 E be the one with the mximl flow, i.e., F f e ¼ mx eef f 0 e. We now prove tht both in the optiml nd the integrl solutions the most congested links crry flows of df f e0e normlized bndwidth units. From Theorem results tht fter the rounding lgorithm the mximl flow of every link e is less thn F f e þ B ¼ F f e þ 1. Since, ech flow contins n integer number of normlized bndwidth units results tht the mximl flow in every link is t most df f e 0e. Now recll tht e0 is bottleneck link in the frctionl solution. Thus it is lower bound on the number of normlized bndwidth units tht re supported by the most congested link in the optiml solution. Since in the optiml solution ech link supports integrl number of normlized bndwidth units, follows tht its most congested link crries flow of t lest df f e0e normlized bndwidth units. This proves tht both the clculted integrl nd optiml solutions hve the sme normlized bndwidth unit, which complete our proof. h

11 60 Y. Bejerno et l. / Computer Networks 51 (007) Theorem. If ll the links hve the sme cpcity C then the bsic lgorithm gurntees -pproximtion rtio. In other words, B i P B =. Proof. We denote by d mx the mximl weight of node nd let e nd l be the links with the mximl flows in the integrl nd the optiml solutions. Clerly both links hve flows of t lest d mx units of the corresponding normlized bndwidth units. From Theorem follows tht F i e < F f e þ d mx 6 F l þ d mx. Since, F l P d mx results tht F i e < F l. Thus, B i ¼ C P C F i F e l ¼ B ; which completes the proof. h.. The bounded weight scheme We now present the bounded weight scheme for networks where the weight d v of very node v V is bounded between d min nd d mx nd we llow rbitrry link cpcities, C e. For such instnces, our scheme ensures tht the clculted normlized bndwidth, B i, is t lest 1/(1 + ) of the optiml solution, where = d mx /d min. Thus, when ll the nodes hve the sme demnd we get -pproximtion rtio. This scheme lso clcultes first frctionl solution nd then uses the rounding method described in Section... Unlike the bsic scheme, this lgorithm uses different frctionl routing formultion, in which only links with enough cpcity crry trffic. We refer to this set of links s the rely group. Consider ny integrl solution nd let B be it normlized bndwidth lloction. In this solution the trffic flows my only trverse over links with cpcity C e P d min B. Thus, for given normlized bndwidth B we define the network rely group to be the set, RðBÞ ¼feje E ^ C e P B d min g: ð7þ We formulte the frctionl routing problem s follows: mx B subject to 8v V ^ d v > 0 : B 6 b v =d v ; 8v V ^ d v ¼ 0 : b v ¼ 0; ð8þ ð9þ 8v V : X unðvþ^ðu;vþrðbþ F u;v ¼ b v þ 8ðu;vÞRðBÞ : F u;v þ F v;u 6 C u;v ; 8ðu;vÞRðBÞ : F u;v P 0; F v;u P 0; 8ðu;vÞ 6 RðBÞ : F u;v ¼ 0; F v;u ¼ 0: X unðvþ^ðu;vþrðbþ F v;u ; ð10þ ð11þ ð1þ ð1þ In this formultion, Constrint (8) ensures tht B lower bound the normlized bndwidth of every node v V with positive weight, while Constrint (9) gurntees tht bndwidth is not llocted to nodes with weight zero. Constrint (10) is the flow conservtion requirement of the rely links. Constrints (11) nd (1) gurntee the cpcity constrints of the rely links, while Constrint (1) ensures tht trffic does not trverse through nonrely links. Unfortuntely, this formultion is not liner problem nd we cnnot simply use the methods described in Section..1, since Constrints (10) (1) depend on the clculted normlized bndwidth B. However, this formultion becomes liner progrm for fixed B. This enble us to find the optiml frctionl B vlue by performing binry serch over B nd checking whether there is frctionl flow solution tht stisfies the predicted normlized bndwidth B. We strt our serch by guessing B ¼ mx ee C e =d mx, which upper bounds the normlized bndwidth. Theorem 5. Consider grph G(V, E) nd bounded weight between d min nd d mx. Then, B i P ð1 þ ÞB, where =d mx /d min. Proof. Let B f be the normlized bndwidth clculted be the frctionl solution. From Theorem nd the definition of rely links RðB f Þ follow tht the integrl flow of ny link e RðB f Þ stisfy the following expression: F i e 6 F f e þ d mxb f 6 1 þ d mx 6 1 þ ; C e C e d min where F i e is the integrl flow through link e before the scling. Thus, the over provisioning rtio k is less thn 1 +. Since, frctionl normlized bndwidth upper bounds the optiml integrl solution, we get tht B i ¼ B f =k P B =ð1 þ Þ nd this completes our proof. h Corollry. Consider grph G(V [ {}, E) where ll nodes hve the sme bndwidth demnd. Then, the bounded weight scheme is -pproximtion lgorithm, i.e., B i P B.

12 Y. Bejerno et l. / Computer Networks 51 (007) The generl scheme Now we describe our generl lgorithm for rbitrry weights nd link cpcities. We construct 5-pproximtion lgorithm by consolidting the rely group method used in Section. nd scling technique like the one used in [8]. Consider grph G(V [ {},E) nd let d mx be the mximl node weight. We divide the nodes with positive weights into disjoint groups D k, k > 0, depending on their weights. A node v V with d v > 0 is included in group k if nd only if dmx < d k v 6 dmx. Thus, k 1 D k ¼ vjv V ^ dmx k < d v 6 d mx k 1 : Recll tht the number of sets is bounded by jvj, when ignoring empty sets. Our scheme clcultes, simultneously, frctionl solution for ll the weight groups tht is bsed on rely group pproch. Then, it uses the rounding lgorithm described in Section.., for obtining integrl flows for ech group D k seprtely. Consider ny integrl solution with normlized bndwidth B, link e E my serve s rely for the flow d v B of ny node v D k only if C e P d v B. Thus, for given normlized bndwidth B, we define group of possible rely links, R k ðbþ, for ech set D k, R k ðbþ ¼ eje E ^ C e P d mx B : k Next, we formulte frctionl routing problem tht llows only links in R k (T) to forwrd trffic of flows in D k. Let denote by F u,v,k the mount of trffic tht trverse link (u,v) for serving the nodes in D k. For generlity, let B v;k ¼ B v if v D k,or0 otherwise. So, the frctionl routing problem is formulted s follows: mx B subject to 8k > 0; v V ^ v D k : B 6 b v;k =d v ; 8k > 0; v V ^ v 6 D k : b v;k ¼ 0; X 8k > 0; v V : F u;v;k ¼ b v;k unðvþ^ðu;vþr k ðbþ þ X unðvþ^ðu;vþr k ðbþ F v;u;k ; ð1þ ð15þ ð16þ 8e ¼ðu; vþ E : X k>0 ðf u;v;k þ F v;u;k Þ 6 C e ; ð17þ 8k > 0; ðu; vþ R k ðbþ : F u;v;k P 0; F v;u;k P 0; ð18þ 8k > 0; ðu; vþ 6 R k ðbþ : F u;v;k ¼ 0; F v;u;k ¼ 0: ð19þ In this formultion, Constrint (1) ensures tht B lower bounds the normlized bndwidth of every node v V. While, Constrints (15) nd (16) re flow conservtion requirements for every weight group D k nd every set R k ðbþ of rely links. Constrints (17) nd (18) ensure the cpcity constrints. Finlly, for the ske of completeness, Constrint (17) ensures tht trffic of node v D k in ny weight group does not trverse ny link e 6 R k ðbþ. Like the formultion of the bounded demnd problem, this is not liner progrm, but it becomes one for fixed B. So, we find the optiml solution by performing binry serch over B nd checking whether there is suitble flow ssignment tht stisfies the predicted normlized bndwidth. Then, we round the flow of ech demnd group D k seprtely nd the finl solution is the collection of routes of ech node v V. Theorem 6. The generl scheme is 5-pproximtion lgorithm for the route selection problem for ny grph G(V [ {}, E) with rbitrry node weights nd link cpcities. Proof. We denote by j e the miniml index of ny rely group tht contins link e E. In other words, j e is the miniml index k such tht C e P dmx B f. k According to Theorem, for every link e E nd ech weight group D k follows tht F i e;k 6 F f e;k þ dmx B f. Where F f k e;k nd F i e;k re the frctionl flow nd integrl flow before the scling originted by nodes in D k nd trverse through link e E. Thus, F i e 6 X F f e;k þ X d mx B f 6 F f k>0 kpj e k e þ d mx B f 6 5C j e e : Accordingly, the over provisioning rtio k is less thn 5. Since, frctionl normlized bndwidth upper bounds the optiml integrl solution, we get tht B i ¼ B f =k P B =5 nd this completes our proof. h.5. User flow routing lgorithm The solution for the user flow model is bsed on the bounded weight scheme presented in Section

13 6 Y. Bejerno et l. / Computer Networks 51 (007) The lgorithm uses similr liner progrm formultion for clculting the frctionl solution when the rely set contins ll the link with cpcity equl of greter thn the predicted normlized bndwidth, i.e., RðBÞ ¼feje E ^ C e P Bg: ð0þ Unlike the ggregted flow model, we would like to clculte in the rounding phse d v routes from ech node v, where ll the routes hve the sme bndwidth lloction equl to the clculted normlized bndwidth. To this end, we modify the rounding lgorithm presented in Section.. s follows. Recll tht node weight d v indictes the number of its ssocited users. Rther thn ssocited single token t v to ech node v, we ssign d v tokens to ech node v V, where ech token represents bndwidth lloction equl to the frctionl normlized bndwidth B f. Beside the initil token ssignment, we utilize the sme rounding method presented bove. Finlly, fter determining the user individul routes we scle down the normlized bndwidth to meet the cpcity constrints. Consequently, from Theorem nd by deploying similr rguments to the ones used in the proof of Theorem we conclude, Corollry. If ll the links hve the sme cpcity C then user flow lgorithm finds the optiml solution. Note tht in the cse of Corollry, the rely set is either empty or contins ll links. In ddition, by minor modifictions of the proof of Theorem 5 it is esy to show, Corollry. When the links hve rbitrry cpcities then the user flow lgorithm gurntees -pproximtion rtio. the scheme presented in [1]. The performnce comprison metric is the miniml per-user bndwidth, B (i.e., miniml b v /d v mong ll nodes). In the following we present typicl results if our simultions. We consider prcticl mesh routers (such s Bel- Air 00) tht re ssocited with bckhul rdio interfces. To simplify the needs for selecting strongly connected mesh topologies with node degrees t most four, we consider grid-like mesh topologies tht coincide squres of sizes nd 15 15, ech one with single gtewy. We first simulted cse tht the gtewy node is locted t the center of mesh nd the users re rndomly plced over the topology, while t most one user cn be ssocited with ech node (i.e., d v = 1). The cpcity of the links between djcent nodes is set to 10 Mbps (i.e., C u,v = 10), while the cpcity of the link between the gtewy nd the infrstructure is set to higher thn the sum of the cpcity of its links to neighbor nodes. We lso ssume tht the cpcity of the ccess chnnels is higher tht the wireless link cpcity between the mesh routers. Fig. 5 presents the simultion result of this cse. The X-xis of the grph represents the number of nodes tht hve n ssocited user. Ech point is obtined through 00 runs nd the results of lightly lod conditions re not plotted for higher redbility. The simultion results indictes tht the INT solution performs very close to the FRAC solution (prticulrly when the network is hevily loded), nd clerly outperforms the heuristic methods. Recll tht the FRAC solution is performnce upper bound nd is gurnteed to be better thn or equl to the optiml integrl solution. In our simultions we did not tke the potentil gin 5. Simultion results We evlute the performnce of the proposed scheme vi simultions. We compre the performnce of the bsic Integrl solution (INT) described in Section. with the performnce of the Frctionl solution (FRAC) nd two heuristic lgorithms: plin Shortest-Pth (SP) lgorithm nd Shortest-Pth Lod-Blncing (SP + LB) lgorithm. In the SP + LB lgorithm, the lest-congested shortest pth is chosen in similr wy 5 to 5 Essentilly, ech edge of the grph mintins weight representing the current lod level of the edge, nd pth with smllest weight sum is chosen. Miniml per-user bndwidth (Mbps) FRAC INT SP+LB SP Number of nodes with users Fig. 5. Miniml per-user bndwidth comprison in mesh with center-locted gtewy node (C u,v = 10, d v = 1).

14 of using the compression schemes in the INT solution, which my in effect boost the performnce of the INT solution over the FRAC (multi-pth) solution. We further show in Fig. 7 comprison of the INT solution with the heuristic methods on the efficiency of the computed routes (i.e., the verge length of the routes). The comprison shows tht the INT method genertes ner-optiml routes in terms of the verge pth length (i.e., very close to the SP method), while the SP + LB method suffers significnt extension of the pth lengths. Tht is, the SP + LB method chieves higher performnce thn the SP method t the expense of longer routes. In contrst the INT solution chieves ner-optiml performnce without such downside, which enbles it to support dely sensitive ppliction like voice more efficiently. Fig. 6 plots the simultion results when the gtewy node is locted ner the corner of the grid topology. As compred to the cse of the centerlocted gtewy node, the performnce gp between the proposed INT solution nd other heuristics widens significntly when the gtewy node is locted ner the corner. It is becuse in ltter cse the overll routing ptterns become more bised nd the routing spce tht the proposed scheme cn exploit increses. The verge pth length comprison (Fig. 8) indictes tht big gp remins between the INT solution nd the SP + LB method, while the length of the pths generted by the INT solution is slightly longer thn tht of the SP method. Now, we turn to the cse tht the link cpcity nd the number of users ssocited with ech node re not fixed. The cpcity of ech link is rndomly chosen between 5 Mbps nd 15 Mbps (i.e., C u,v = 5 15). The users re rndomly plced so tht up to five users re ssocited with ech node (i.e., Miniml per-user bndwidth (Mbps) FRAC INT SP+LB SP Number of nodes with users Fig. 6. Miniml per-user bndwidth comprison in mesh with corner-locted gtewy node (C u,v = 10, d v = 1). Y. Bejerno et l. / Computer Networks 51 (007) Averge pth length (hops) SP+LB INT SP Number of nodes with users Fig. 7. Averge pth length comprison in mesh with center-locted gtewy node (C u,v = 10, d v = 1). Averge pth length (hops) SP+LB INT SP Number of nodes with users Fig. 8. Averge pth length comprison in mesh with corner-locted gtewy node (C u,v = 10, d v = 1). d v = 1 5). Insted of the more sophisticted methods described in Sections. nd., we continue to use the bsic Integrl solution (INT) described in Section. to demonstrte tht even the simple bsic method performs firly well in generl cses. Figs. 9 nd 10 depict the cses of center-locted gtewy node nd corner-locted gtewy node, respectively. As expected, the gp between the FRAC solution nd the INT solution increses s compred with the cse of fixed C u,v nd d v, but other thn tht the sme generl trend s in Figs. 5 nd 6 cn be observed. To exmine the impct of the network size, we lso simulte mesh. The results re presented in Figs. 11 nd 1. As the number of users increses, the rounding error shrinks nd the gp

15 6 Y. Bejerno et l. / Computer Networks 51 (007) Miniml per-user bndwidth (Mbps) FRAC INT SP+LB SP Miniml per-user bndwidth (Mbps) FRAC INT SP+LB SP Number of nodes with users Number of nodes with users Fig. 9. Miniml per-user bndwidth comprison in mesh with center-locted gtewy node (C u,v = 5 15, d v = 1 5). Fig. 11. Miniml per-user bndwidth comprison in mesh with center-locted gtewy node (C u,v = 5 15, d v = 1 5). Miniml per-user bndwidth (Mbps) FRAC INT SP+LB SP Miniml per-user bndwidth (Mbps) FRAC INT SP+LB SP Number of nodes with users Number of nodes with users Fig. 10. Miniml per-user bndwidth comprison in mesh with corner-locted gtewy node (C u,v = 5 15, d v = 1 5). Fig. 1. Miniml per-user bndwidth comprison in mesh with corner-locted gtewy node (C u,v = 5 15, d v = 1 5). between the INT solution nd the FRAC solution decreses. Finlly, let s exmine how much performnce gin cn be chieved by using the user-flow routing lgorithm described in Section.5. Since the frctionl solution of this lgorithm is the sme s tht of the ggregted flow lgorithm, the performnce of the user-flow solution must locte between the FRAC solution nd the INT solution (for the ske of clrity we omit these curves from the chrts). The performnce gp between the FRAC nd INT solutions is very smll in ll simultions cses presented, the performnce gin by the user-flow method is miniml. To conclude, our simultion results reconfirm the rgument mde by Gnjli nd Keshvrzin in [17] tht the performnce gin of multi-pth routing is not big. 6. Conclusion In this pper, we focused on determining the routes nd bndwidth lloction of the trffic flows of WMNs for mximizing the fir shre llocted to the users. We presented polynomil-time lgorithms for this purpose nd nlyzed the qulity of our solutions ginst the optiml solutions. The simultions show tht our lgorithms, indeed, find neroptiml solutions. These lgorithms cn be used under the centrlized mngement rchitecture, in which NOC locted in the wired network performs

16 Y. Bejerno et l. / Computer Networks 51 (007) network mngement functions for WMNs. The bsic ide of our lgorithms is not limited to the current LP problem formultions, but cn rther be pplied to vrious other problem formultions, for instnce, ones with different interference models. References [1] R. Bruno, M. Conti, E. Gregori, Mesh networks: commodity multihop d hoc networks, IEEE Commun. Mg. 7 () (005) [] R. Drves, J. Pdhye, B. Zill, Routing in multi-rdio, multihop wireless mesh networks, in: Proceedings of ACM Mobicom 0, Phildelphi, PA, USA, September 00, pp [] A. Rniwl, T.-C. Chiueh, Architecture nd lgorithms for n IEEE bsed multi-chnnel wireless mesh network. in: Proceedings of IEEE INFOCOM 05, Mimi, FL, USA, Mrch 005. [] R. Krrer, A. Sbhrwl, E. Knightly, Enbling lrge-scle wireless brodbnd: the cse for TAPs, in: Proceedings of HotNets 00, Cmbridge, MA, USA, November 00. [5] I.F. Akyildiz, X. Wng, W. Wng, Wireless mesh networks: survey, Computer Networks 7 () (005) [6] S. 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Johnson, Computers nd Intrctbility: A Guide to the Theory of NP-completeness, Freemn Publiction, New York, [6] G. Zussmn, A. Segll, Energy efficient routing in d hoc disster recovery networks, in: Proceedings of IEEE INFO- COM 0, Sun Frncisco, CA, USA, 00. [7] N.Grg, J. Kemnn, Fster nd simpler lgorithms for multicommodity flow nd other frctionl pcking problems, in: Proceedings of the 9th Annul IEEE Computer Society Conference on Foundtions of Computer Science, FOCS 98, Plo Alto, CA, USA, November 1998, pp [8] S.G. Kolliopoulos, C. Stein, Improved pproximtion lgorithms for unsplittble flow problems, in: Proceedings of the 8th Annul IEEE Computer Society Conference on Foundtions of Computer Science, FOCS 97, Mimi Bech, FL, USA, October 1997, pp Yigl Bejerno received the B.Sc. degree in computer engineering (summ cum lude), the M.Sc. degree in computer science, nd the Ph.D. degree in electricl engineering from the Technion Isrel Institute of Technology, Hif, Isrel, in 1991, 1995, nd 000, respectively. He is currently Member of the Technicl Stff with the Networking Reserch Center, Bell Lbortories, Lucent Technologies, Murry Hill, NJ. His reserch interests re minly mngement spects of high-speed nd wireless networks, including the res of mobility mngement, network monitoring, topology discovery, qulity of service (QoS) routing, wireless LAN, nd wireless mesh networks.

17 66 Y. Bejerno et l. / Computer Networks 51 (007) Seung-Je Hn received the B.S. nd M.S. degrees in computer engineering from Seoul Ntionl University, Seoul, Kore, nd the Ph.D. degree in Computer Science nd Engineering from the University of Michign, Ann Arbor. He ws Member of Technicl Stff of the Wireless Reserch Lbortory, Bell Lbortories, Lucent Technologies, Murry Hill, NJ. Currently he is n ssocite professor in the computer science deprtment of Yonsei University, Seoul, Kore. His reserch interests include mobile networking, wireless Internet, nd network mngement. Amit Kumr received the B.S. degree from Indin Institute of Technology, Knpur, Indi, nd the Ph.D. degree from Cornell University. He ws Member of Technicl Stff in Bell Lbortories, Lucent Technologies, Murry Hill, NJ. Currently he is n ssistnt professor in the computer science deprtment of Indin Institute of Technology, New Delhi, Indi.