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1 Computer Networks 53 (29) Contents lists ville t ScienceDirect Computer Networks journl homepge: Congestion-wre chnnel ssignment for multi-chnnel wireless mesh networks q A. Hmed Mohsenin Rd, Vincent W.S. Wong * Deprtment of Electricl nd Computer Engineering, The University of British Columi, Vncouver, Cnd rticle info strct Article history: Received 1 July 28 Received in revised form 21 Jnury 29 Accepted 11 My 29 Aville online 23 My 29 Responsile Editor: G. Morito Keywords: Multi-chnnel wireless mesh networks Distriuted lgorithm Chnnel ssignment Congestion control Cross-lyer design Frequency overlpping In this pper, we propose distriuted congestion-wre chnnel ssignment (DCACA) lgorithm for multi-chnnel wireless mesh networks (MC WMNs). The frequency chnnels re ssigned ccording to the congestion mesures which indicte the congestion sttus t ech link. Depending on the selected congestion mesure (e.g., queueing dely, pcket loss proility, nd differentil cklog), vrious design ojectives cn e chieved. Our proposed distriuted lgorithm is simple to implement s it only requires ech node to perform locl serch. Unlike most of the previous chnnel ssignment schemes, our proposed lgorithm ssigns not only the non-overlpped (i.e., orthogonl) frequency chnnels, ut lso the prtilly-overlpped chnnels. In this regrd, we introduce the chnnel overlpping nd mutul interference mtrices which model the frequency overlpping mong different chnnels. Simultion results show tht in the presence of elstic trffic (e.g., TCP Vegs or TCP Reno) sources, our proposed DCACA lgorithm increses the ggregte throughput nd lso decreses the verge pcket round-trip compred with the previously proposed Lod-Awre chnnel ssignment lgorithm. Furthermore, in congested IEEE network setting, compred with the use of three non-overlpped chnnels, the ggregte network throughput cn further e incresed y 25% nd the verge round-trip time cn e reduced y more thn one hlf when ll the 11 prtilly-overlpped chnnels re used. Ó 29 Elsevier B.V. All rights reserved. 1. Introduction Wireless mesh networks (WMNs) hve recently received n incresing ttention to provide uiquitous nd inexpensive lst-mile Internet ccess. A WMN consists of numer of sttionry wireless mesh routers, forming wireless ckone. These routers serve s ccess points for vrious wireless moile devices. Some of the routers lso ct s gtewys to the Internet vi high-speed wired links. Moile devices first trnsfer dt to their ssocited q This work ws supported y the Nturl Sciences nd Engineering Reserch Council of Cnd under Grnt Numer This pper ws presented in prt t the IEEE ICC, Glsgow, Scotlnd, UK, June 27. * Corresponding uthor. Tel.: E-mil ddresses: hmed@ece.uc.c (A.H. Mohsenin Rd), vincentw@ ece.uc.c (V.W.S. Wong). router, nd the dt is then trnsferred to the Internet vi the intermedite routers in multi-hop mnner [1,2]. The ggregte cpcity nd the performnce of WMNs cn e incresed y the use of multiple chnnels [3]. In this scenrio, ech wireless mesh router is equipped with multiple network interfce crds (NICs). Ech NIC opertes on distinct frequency chnnel in the IEEE 82.11//g nds. Two neighoring routers cn communicte with ech other if ech one hs n NIC operting on the sme frequency chnnel. Within the IEEE frequency nds, the numer of ville chnnels is limited. For exmple, the IEEE 82.11/g stndrds hve 11 chnnels, of which three chnnels re non-overlpped. The numer of operting chnnels in the IEEE stndrd is 79, of which 12 chnnels re non-overlpped. These imply tht some logicl links my e ssigned to the sme chnnel. Interference will occur if these links re close to ech other /$ - see front mtter Ó 29 Elsevier B.V. All rights reserved. doi:1.116/j.comnet

2 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Interference mong neighoring links cn reduce their effective dt rte nd potentilly cuse network congestion. For pplictions which use TCP (trnsmission control protocol) in the trnsport-lyer, if the links ecome congested, there will e reduction of the ggregte throughput s well s their qulity-of-service. Thus, efficient chnnel ssignment is crucil to reduce interference mong neighoring trnsmissions. There exists wide rnge of proposed chnnel ssignment lgorithms for multi-chnnel wireless mesh networks (MC WMNs) in the literture. One pproch is to formulte chnnel ssignment prolem s n optimiztion prolem [4 18]. Ds et l. [5] proposed n optimiztionsed lgorithm tht mximizes the numer of logicl links tht cn e ctive simultneously, suject to interference constrints. Chen et l. [6] devised chnnel ssignment strtegy which ssigns the chnnels in order to lnce the trffic lod etween different chnnels. The lgorithms in [5 8] re sttic nd ssign the chnnels permnently. There lso exist some dynmic lgorithms which updte the ssigned chnnels either in short-term sis (e.g., pcket-y-pcket [9 11]) or long-term sis (e.g., every severl minutes or hours [12,13,15,16,14]). Short-term chnnel updtes require fst chnnel switching which cn e chllenge in the existing commercil IEEE interfces with switching ltency in the order of 1 ms [19,2]. Another chllenge is the required fst coordintion to ensure tht the sending nd receiving routers use the sme chnnel. On the other hnd, long-term intervl chnnel updtes do not require fst switching nd coordintion. They cn lso use the existing IEEE commodities. Rniwl nd Chiueh [15] proposed long-term dynmic chnnel ssignment lgorithm clled Lod-Awre lgorithm. By monitoring the mount of trffic eing trnsmitted over ech frequency chnnel, wireless mesh routers ssign their NICs with those chnnels which hve minimum usge within their neighorhood. In [12], Alicherry et l. proposed n interference-free scheduler which ims in mximizing the ndwidth llocted to ech router suject to the constrint tht the llocted ndwidth is in proportion to its ggregte trffic demnd. Kodilm nd Nndgopl [13] lso proposed n lgorithm tht mximizes the network throughput suject to the minimum rte requirements for ech flow. It hs een lso recently shown in two independent works in [21,22] tht using prtilly-overlpped frequency chnnels cn further improve network performnce. Lst ut not lest, the study of chnnel ssignment when smrt directed ntenns re used is presented in [23]. Non-coopertive chnnel ssignment is lso studied in [24]. In summry, most of the previous chnnel ssignment lgorithms mentioned ove hve one or more of the following performnce ottlenecks: First, mny of these lgorithms re centrlized. They require strong coordintion nd result in high computtionl complexity nd significnt signlling overhed. Second, they only tke into ccount the orthogonl (i.e., non-overlpped) frequency chnnels, ut not the prtilly overlpped chnnels. Thus, the frequency spectrum is not utilized efficiently. Third, some of these lgorithms re sttic. They cnnot dpt to the time-vrying fetures of the networks such s the vrile trffic demnds. Fourth, most of the previous lgorithms re sed on vrious heuristic design. This my e due to the lck of ccurte cpcity models in terms of the chnnel ssignment vriles. Finlly, none of the lgorithms mentioned ove tke congestion informtion into ccount. In this pper, we propose the distriuted congestionwre chnnel ssignment (DCACA) lgorithm which overcomes the ove performnce ottlenecks in ll five spects. Our proposed lgorithm is distriuted nd is executed y ech wireless mesh router in n synchronous mnner. In this regrd, ech node only needs to perform simple locl serch to dequtely ssign the frequency chnnels to suset of logicl links. Moreover, our proposed lgorithm is le to ssign not only the non-overlpped chnnels, ut lso ll the ville prtilly-overlpped chnnels with ritrry overlpping portions. In this regrd, we propose two key mtrices, clled chnnel overlpping mtrix nd mutul interference mtrix, tht re le to mthemticlly model the frequency overlpping mong the chnnels. Our proposed lgorithm is longterm dynmic chnnel ssignment lgorithm. It ssigns the frequency chnnels sed on the most recent congestion informtion mesured cross the network. Unlike other distriuted chnnel ssignment lgorithms which suggest selfish ctions y ech wireless mesh router, our proposed lgorithm is sed on coopertion mong the routers. This is indeed necessry for chieving the optiml network performnce in distriuted fshion. Finlly, our lgorithm is designed to solve mthemticlly formulted congestion-wre chnnel ssignment prolem in the generl form of mximizing weighted summtion of link cpcities. Depending on the selected weighting prmeters, solving our formulted prolem results in chieving some well-known resource lloction design ojectives for oth fixed nd elstic trffic ptterns. In this regrd, we derive closed-form cpcity model for ech logicl link in terms of our defined chnnel ssignment vriles s well s the chnnel overlpping nd mutul interference mtrices. Simultion results show tht if TCP Vegs is used, then our proposed DCACA lgorithm increses the ggregte throughput y 11.5% nd decreses the verge pcket round-trip time y 35.3% compred to the Lod-Awre lgorithm [15]. On the other hnd, if TCP Reno is used, then DCACA lgorithm increses the ggregte throughput y 9.8% nd decreses the verge pcket round-trip time y 28.7% compred to the Lod-Awre lgorithm [15]. Furthermore, in congested IEEE wireless network setting, compred with the use of 3 non-overlpped frequency chnnels, the ggregte throughput cn further e incresed y 25% nd the verge round-trip time cn e reduced y more thn one hlf when ll the 11 prtilly-overlpped chnnels re used. The rest of this pper is orgnized s follows. The prolem formultion is descried in Section 2. Our proposed link cpcity models re developed in Section 3. In Section 4, we propose the DCACA lgorithm nd provide the proof of its convergence. Performnce ssessments nd comprison studies re presented in Section 5. Concluding remrks re given in Section 6. A list of the key nottions tht we used in this pper is shown in Tle 1.

3 254 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Tle 1 Index of key nottions N; N Set of nodes (i.e., wireless mesh routers) nd the numer of nodes, respectively L; L Set of links nd the numer of links, respectively C; C Set of chnnels nd the numer of chnnels, respectively I n; I n Set of NICs in node n nd the numer of NICs in node n, respectively I Mximum numer of NICs mong ll nodes L in n ; Lout n Set of ll incoming nd outgoing links of node n, respectively L n; L n;i Set of links of node n nd set of links using NIC i on node n, respectively x l Chnnel ssignment vector corresponding to link l x Chnnel ssignment vector corresponding to ll links Set of ll fesile chnnel ssignment vectors q l ; k l Congestion mesure of link l nd the persistent proility of link l, respectively k; q Congestion mesure of ll links nd the persistent proility of ll links, respectively c l Cpcity of link l T G Time intervl t which prolem (CACA) needs to e solved T L Time intervl t which prolem (LOCAL-CACA) is eing solved c l ; c min SINR for link l nd minimum required SINR, respectively T S Symol period K A constnt which depends on the modultion scheme F u Power spectrl density function of the nd-pss filter for chnnel u W Chnnel overlpping mtrix w uv Entry in the uth row nd the vth column of chnnel overlpping mtrix W M lk Mutul interference mtrix corresponding to links l nd k g lk Pth loss from the trnsmitter node of link l to the trnsmitter node of link k e lk Euclidin distnce etween the trnsmitter node of link l nd the trnsmitter node of link k j Pth loss exponent p l Trnsmission power of the trnsmitter node of link l g l Noise power t the receiver node of link l d Roll-off fctor A constnt which depends on the ntenn gins nd signl wvelengths O l ; O l Opponent set of link l nd the numer of opponent links of link l, respectively Hdmrd product s l The node which is responsile for chnnel ssignment of link l x n Chnnel ssignment vectors corresponding to ll links other thn links of node n L mx Mximum links connected to ny node in the network r sd Dt rte for the end-to-end trffic from source node s to destintion node d T Set of ll time slots T G Set of ll time slots t which prolem (CACA) needs to e solved T L;n Set of ll time slots t which prolem (LOCAL-CACA) is eing solved y node n wðtþ The ojective function of prolem (CACA) t time slot t 2. Prolem formultion Consider n MC WMN with N s the set of wireless nodes (i.e., wireless mesh routers) nd L s the set of unidirectionl logicl links 1. We define N ¼jNj nd L ¼jLj s the crdinlity of set N nd L, respectively. For ech node n 2 N, let L out n denote the set of ll outgoing links from node 1 Here we ssume tht ll logicl links in set L re used for pcket trnsmissions. If there is logicl link tht does not elong to ny of the routing pths in the network, we simply ssume tht it does not exist. n nd L in n define L n ¼ L out n denote the set of ll incoming links to node n. We [ L in n. Thus, [ n2nl n ¼ L. Let C denote the set of ll ville frequency chnnels nd C ¼jCj denote the crdinlity of set C. For ech node n 2 N, we lso define I n s the set of its NICs. The crdinlity of set I n is denoted y I n. We hve I ¼ mx n2n I n. The logicl topology is ssumed to e symmetric. Tht is, if l 2 L out n \ L in m is logicl link in the direction from node n to node m, then there exists nother link k 2 L in n \ Lout m in the direction from node m to node n. The logicl topology is lso ssumed to e ripple-effect free [16,15,17]. Two smple ripple-effect free MC WMN logicl topologies re Hycinth [15] nd TiMesh [17]. They re shown in Fig. 1 nd, respectively. In ripple-effect free logicl topology, there exists n exclusive (i.e., not shred) NIC in t lest one end of ech logicl link. This limits the chnnel dependency mong the links. Hence, ssigning new chnnel to one link does not trigger series of chnnel re-ssignments cross the network (see Fig. 3 in [15]). Thus, distriuted chnnel ssignment is fesile. For ech node n 2 N nd ny of its NICs i 2 I n, we define L n;i s the set of links tht use NIC i in node n. Notice tht [ i2in L n;i ¼ L n. Considering the MC WMN logicl topology in Fig. 1, L out ¼fl 1 ; l 3 g; L in ¼fl 2; l 4 g; L ¼fl 1 ; l 2 ; l 3 ; l 4 g; L ;1 ¼fl 1 ; l 2 g, nd L ;2 ¼fl 3 ; l 4 g. We lso hve: L out c ¼ fl 2 ; l 5 ; l 7 g; L in c ¼fl 1 ; l 6 ; l 8 g; L c ¼fl 1 ; l 2 ; l 5 ; l 6 ; l 7 ; l 8 g; L c;1 ¼fl 1 ; l 2 g, nd L c;2 ¼ fl 5 ; l 6 ; l 7 ; l 8 g. On the other hnd, in Fig. 1, L out d ¼fl 4 ; l 9 ; l 11 ; l 14 g; L in d ¼fl 3; l 1 ; l 12 ; l 13 g; L d ¼fl 3 ; l 4 ; l 9 ; l 1 ; l 11 ; l 12 ; l 13 ; l 14 g; L d;1 ¼fl 3 ; l 4 ; l 9 ; l 1 g, nd L d;2 ¼fl 11 ; l 12 ; l 13 ; l 14 g. For ech logicl link l 2 L, we define C 1 inry chnnel ssignment vector x l. The ith entry of x l is equl to 1 if chnnel i is ssigned to link l; otherwise, it is equl to. For exmple, if C ¼ 4 nd the third chnnel is ssigned to logicl link l, then x l ¼ ½ 1 Š T. Since one frequency chnnel is ssigned to ech logicl link, one of the entries of x l should e equl to 1 nd the rest should e. This requires tht: 1 T x l ¼ 1; 8l 2 L; ð1þ where 1 denotes C 1 vector with ll entries equl to 1. From (1), it is cler tht for ny two ritrry links l; k 2 L, if they operte over the sme chnnel, then x Tx l k ¼ 1; otherwise, x T l x k ¼. On the other hnd, since ll links which shre the sme NIC need to use the sme frequency chnnel, for ech wireless node n 2 N nd ny of its NICs i 2 I n, we hve: x l ¼ x k ; 8l; k 2 L n;i : ð2þ For the simplicity of exposition, we stck up the chnnel ssignment vectors corresponding to ll links nd denote the otined LC 1 vector y x. In this pper, we re interested in finding x to solve the following glol congestionwre chnnel ssignment (CACA) prolem: mximize k l c l ðxþ; ðcacaþ x2 where l2l ¼fx : x 2f; 1g LC ; 1 T x l ¼ 1; x l ¼ x k ; 8n 2 N; i 2 I n ; l; k 2 L n;i g: Here f; 1g LC denotes the set of ll LC 1 inry vectors nd denotes the set of ll fesile chnnel ssignment vectors. ð3þ

4 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Fig. 1. Two smple ripple-effect free MC WMN topologies with 1 wireless mesh routers nd 25 wireless mesh clients. The nottions c l ðxþ nd k l denote the cpcity nd congestion mesure of logicl link l 2 L, respectively. The more the link l is congested, the higher the vlue of k l will e. By solving prolem (CACA), the frequency chnnels re ssigned to mximize weighted summtion of link cpcities where the congestion mesures ct s weights. In this regrd, the congested logicl links re provided with more cpcities. Vrious congestion mesures cn e considered. For exmple, following the steps in [22], we cn show tht the network utility mximiztion (NUM) prolem [25 28] in the presence of elstic trffic sources cn e reduced to solving prolem (CACA) if k l denotes the congestion price on link l. In this cse, the congestion prices depend on the trnsport-lyer protocol is eing used. For exmple, the congestion prices re queueing dely nd pcket loss proility for TCP Vegs [29] nd TCP Reno [3], respectively. On the other hnd, we my choose k l to e the differentil cklog corresponding to logicl link l 2 L. Tht is, the difference etween the queue cklogs in the trnsmitter nd receiver nodes of logicl link l. The differentil cklog is n indiction of reltive congestion. In this cse, solving prolem (CACA) results in finding the optiml solution of the mximum weight mtching (MWM) prolem [31,32], which stilizes the constrined queueing systems nd leds to mximum ggregte network throughput. For the simplicity of exposition, we ssume tht time is divided into equllength slots T ¼f; 1; 2;...g. In prctice, regrdless of the selected congestion mesures, we re interested in solving prolem (CACA) periodiclly, e.g., every T G time slots. This is ecuse the congestion mesures re usully time-vrying. Intervl prmeter T G cn e in the order of severl seconds, couple of minutes, or few hours depending on the selected congestion mesures. Let T G T denote the set of ll time slots t which prolem (CACA) needs to e solved.

5 256 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Any two consecutive memers of set T G should e exctly T G time units wy from ech other. For exmple, T G ¼f1; 1 þ T G ; 1 þ 2T G ; 1 þ 3T G ;...g. 1 () δ = 1 3. Link cpcity s function of chnnel ssignment vector PSD.5 In n MC WMN, the cpcity of logicl link is function of severl prmeters including the trnsmission powers, node positions, nd the ssigned frequency chnnels. Since the nodes re sttionry, their positions re known in dvnce. In this pper, we limit our study to the fixed trnsmission powers. Thus, the link cpcity is only function of the chnnel ssignment vector x. The informtiontheoretic cpcity of the logicl link l 2 L cn e expressed s [33]: c l ðxþ ¼ 1 T S logð1 þ Kc l ðxþþ; where c l ðxþ denotes the signl to interference nd noise rtio (SINR) of link l; T S is the symol period, nd K is constnt which depends on the modultion scheme eing used. Next, we show tht the vlue of c l ðxþ cn e determined in the presence of oth non-overlpped nd prtilly-overlpped frequency chnnels. Assume tht u nd v re two ville chnnels within the IEEE frequency nd (i.e., u; v 2 C). Let F u ðxþ nd F v ðxþ denote the trnsfer functions of the nd-pss filters for frequency chnnels u nd v, respectively. The PSD functions cn e otined from the chnnels frequency responses. Without loss of generlity, we ssume the use of rised cosine filters [33]. An importnt prmeter to identify the frequency response of this filter is the rolloff fctor d. Fig. 2 shows the response of the IEEE chnnels with d equls to 1 nd.25, respectively. We cn see tht the lower the roll-off fctor, the smller is the overlpping portion mong the neighoring chnnels. To model the overlpping mong different chnnels, we define symmetric C C chnnel overlpping mtrix W. The entry in the uth row nd the vth column of W is denoted y sclr w uv nd is defined s follows: R 1 1 w uv ¼ F uðxþf v ðxþdx ð4þ R 1 : ð5þ 1 F2 uðxþdx Let p l denote the trnsmission power of the trnsmitter node of link l. Also, let g lk denote the pth loss from the trnsmitter node of link l to the receiver node of link k. Assuming tht oth links l; k re ctive, the interference power from link l on link k cn e modeled s: x T l Wx kg lk p l ¼ w uv g lk p l : 3.1. All-t-once scheduling We first consider the ll-t-once scheduling model. In this model, ll the links cn e ctive simultneously nd there is no crrier sensing mechnism in the MAC protocol. From (6), c l ðxþ ¼ðg ll p l Þ= ð6þ x T k Wx lg kl p k þ g l!; 8 l 2 L; ð7þ PSD where g l denotes the therml noise power t the receiver node of link l. Replcing (7) in (4), we cn otin the cpcity model in the ll-t-once scheduling scenrio s: c l ðxþ ¼ 1 T S log Kg 1 þ ll p P l!; xtwx 8 l 2 L: k lg kl p k þ g l From (8), prolem (CACA) ecomes: mximize x2 log Frequency (MHz) k l T l2l S ð8þ Kg 1 þ ll p P l!: xtwx ð9þ k lg kl p k þ g l Notice tht 1=T S is multiplied to ll the terms in the summtion. Thus, the vlue of the symol period T S does not ffect the solution of the chnnel ssignment prolem in (9). On the other hnd, in most prcticl cses, the multipliction fctor K is lrge [34]. For exmple, in the IS-95 CDMA which is indeed direct-sequence spred-spectrum system, the processing gin is 128 [34, Chpter 3.4.3, p. 91]. As result, logð1 þ Kc l ðxþþ logðkc l ðxþþ. This high SINR regime pproximtion is widely used in the networking literture (cf. [35 37]). In this cse, the ojective function in prolem (9) cn e written s: k l logðkg ll p l Þ k l log x T k Wx lg kl p k þ g l!: l2l l2l ð1þ Since the first term is independent of the chnnel ssignment vector x, prolem (9) reduces to: minimize x2 k l log l2l () δ =.25 Fig. 2. The ville eleven prtilly-overlpped chnnels in frequency nd for roll-off fctor d ¼ 1 nd d ¼ :25. The numer on ech curve indictes the corresponding chnnel numer. Chnnels 1, 6, nd 11 re non-overlpped (orthogonl). x T k Wx lg kl p k þ g l!: ð11þ

6 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Exclusive scheduling Most of the existing MAC protocols do not implement the ll-t-once scheduling. Insted, they use vrious crrier sensing mechnisms. The wireless mesh routers compete to ccess the shred medium. In this cse, only one logicl link within neighorhood cn e ctive t time. Next, we explin how we cn tke the effect of MAC lyer chnnel ccess competition into ccount. In this regrd, we first need to clrify the concept of mutul interference. In n MC WMN, where only the non-overlpped frequency chnnels re eing used, two links l; k 2 L re defined mutully interfered with ech other whenever they re ssigned to the sme chnnel (i.e., x T l x k ¼ 1) nd the sender of one link is within the interference rnge of the receiver of the other link. In this cse, links l nd k cnnot e ctive simultneously. The interference rnge is defined s the region where given receiver cnnot decode the signl correctly if there is nother trnsmission within tht rnge. Given the modultion scheme, the interference rnge depends on the minimum required SINR, which is denoted y c min. Now consider the cse where the frequency chnnels re prtilly overlpped. If the interference power of the trnsmission on link k cuses the SINR on link l to e lower thn c min, then the trnsmitter of link k is within the interference rnge of the receiver node of link l. Tht is, g ll p l w vu g lk p k þ g l < c min : ð12þ Without loss of generlity, we model the pth loss g kl using the Friis free spce model [33]: g kl ¼ ðe kl Þ j ; ð13þ where e kl is the Eucliden distnce etween the trnsmitter node of link k nd the receiver node of link l; j is the pth loss exponent, nd is constnt which depends on the trnsmitter nd the receiver ntenn gins nd signl wvelength. By sustituting (13) into (12) nd rerrnging the terms, link k interferes with link l if sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j p e kl < k w g ll p l =c min g vu : ð14þ l Fig. 3. Different interference rnges depending on the frequency chnnel seprtion ju vj. Logicl links l nd k use chnnels u nd v in frequency nd, respectively. The importnce of (14) is tht we now hve different interference rnges depending on the ssigned chnnels to the neighoring links. The smller the portion of the frequency overlpping, the shorter the interference rnge will e. Given tht the ndwidth nd the roll-off fctor re the sme in ll rised cosine chnnel filters, the interference rnge only depends on the frequency chnnel seprtion ðju vjþ. This fct is illustrted in Fig. 3 where d ¼ 1. The outermost circle indictes the interference rnge of the receiver node d when ju vj ¼ (i.e., the sme chnnel is ssigned to links k nd l). The next circle shows the interference rnge when ju vj ¼1. The innermost circle corresponds to the interference rnge when ju vj ¼3. When ju vj > 3, there is no overlpping etween frequency chnnels m nd n for IEEE (see Fig. 2). Thus, the corresponding interference rnges re equl to zero. Note tht in this exmple, trnsmissions on link l interfere with the trnsmissions on link k only when either ju vj ¼orju vj ¼1. For ny links l; k 2 L, we define symmetric C C mutul interference mtrix M lk. If either sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j p e lk < l ; ð15þ g kk p k =c min g k w uv or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j p e kl < k w g ll p l =c min g vu ; ð16þ l then the entry in the uth row nd the vth column of M lk is equl to 1; otherwise, it is equl to. If the trnsmission powers re fixed, the mutul interference mtrices re constnt for sttionry MC WMNs. For the scenrio in Fig. 3, the corresponding mutul interference mtrices re tridigonl with ll digonl, sudigonl, nd superdigonl entries equl to one: M lk ¼ M kl ¼ : ð17þ Note tht if the logicl links l nd k re fr enough from ech other, then ll entries of M lk ecome zero. According to the definitions of the chnnel ssignment vector nd the mutul interference mtrix, links l nd k cnnot e ctive simultneously if x T l M lk x k ¼ 1. From this, we define the opponent set of link l s follows: O l ðxþ ¼ k : k 2 L nflg; x T l M lk x k ¼ 1 ; 8 l 2 L: ð18þ The crdinlity of the set O l is denoted y O l. Since the mutul interference mtrix M lk is symmetric, link k 2 O l ðxþ, if nd only if link l 2 O k ðxþ. Let q l denote the persistent proility of logicl link l 2 L. Tht is, t ech time slot t 2 T, link l is ctive with proility q l. The persistent proilities cn e otined directly from the MAC protocols which re eing used (cf. [38,39]). Considering

7 258 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) the IEEE DCF, the persistent proilities cn e clculted ccording to the size of the contention windows [4,41]. In generl, regrdless of the employed MAC protocol, ech node n 2 N cn mesure the persistent proility q l for ll of its outgoing links l 2 L out n. For nottion simplicity, we stck up the persistent proilities of ll logicl links nd denote the otined L 1 vector y q. According to the exclusive scheduling model, the trnsmissions on ny logicl link l 2 L is successful if no other link in the opponent set O l is ctive t the sme time. This hppens with proility [42]: Y Y q l ð1 q k Þ¼q l ð1 x T l M lkx k q k Þ; ð19þ k2o l ðxþ where 1 x T l M lkx k q k ¼ 1 q k; if k 2 O l ðxþ; 1; otherwise: ð2þ Logicl link l 2 L nd ny link k 2 L nðo l ðxþ[flgþ do not mutully interfere with ech other. They cn e ctive simultneously. However, we should still consider the effect of their interference power on ech other. In fct, ssuming tht link l is ctive nd no other link k 2 O l ðxþ is ctive, the verge interference power on link l cn e otined s: q k x T k Wx lg kl p k ¼ q k 1 x T k M klx l x T k Wx l gkl p k k2lnðo l ðxþ[flgþ ¼ q k x T k ðð1 M klþwþx l g kl p k ; ð21þ where denotes the Hdmrd product 2 nd 1 is C C mtrix with ll entries equl to 1. The entry in the uth row nd vth column of mtrix ð1 M kl ÞW is equl to w uv if logicl links l nd k re not mutully interfered over chnnels u nd v; otherwise, it is equl to zero. From (4), (19), nd (21), we cn otin the verge link cpcities to e s follows: c l ðxþ¼ q l T S Y log 1 þðkg ll p l Þ= 8l 2 L: 1 x T l M! lkx k q k q k x T k ðð1 M klþwþx l g kl p l þ g l!!; ð22þ As the minimum required SINR tends to zero (i.e., c min! ), for ll links l 2 L, the opponent set O l ecomes n empty set nd q l! 1. Tht is, ll links cn e lwys ctive simultneously. We would lso hve M lk ¼ for ny l 2 L nd ech k 2 L nflg. In fct, the cpcity model in (8) is specil cse of the cpcity model in (22). Similr to (9), we cn replce c l ðxþ in prolem (CACA) with (22) nd otin the complete formultion of our congestionwre chnnel ssignment model. We lso notice tht for 2 The Hdmrd product of two C C mtrices A nd B is C C mtrix whose entry in the uth row nd vth column is equl to the product of the entry in the uth row nd vth column of A nd the entry in the uth row nd vth column of B [43]. ll links l; k 2 L, the mtrices W; M lk, nd ð1 M kl ÞW re constnt nd independent of x. Thus, they cn e otined off-line nd lter e used in the corresponding lgorithm implementtions. Next, we propose distriuted lgorithm to solve the congestion-wre chnnel ssignment prolem (CACA). 4. Distriuted congestion-wre chnnel ssignment (DCACA) lgorithm Since the optimiztion vriles re inry, prolem (CACA) is comintoril prolem nd is NP-hrd [44]. It cn e solved in centrlized mnner. In this regrd, the congestion mesures for ll links (i.e., k ¼ðk l ; 8 l 2 LÞ) need to e gthered every T G time slots (see Section 2) in pre-uthorized node (e.g., one of the gtewys). The pre-uthorized node then solves prolem (CACA) nd nnounces the selected optiml chnnels to ll other nodes. In tht cse, the pre-uthorized node should solve comintoril prolem with C L comintions. This my not e trctle when the network grows in size nd the numer of logicl links increses. In this section, we propose distriuted lgorithm to otin ner optiml solution of prolem (CACA) with low complexity. In this regrd, ech node n 2 N is responsile for ssigning the optiml chnnels only to suset of links. Recll from Section 2 tht we ssume the logicl topology to e ripple-effect free [16,15,17]. Two smple ripple-effect free MC WMN logicl topologies re Hycinth [15] nd TiMesh [17], which re shown in Fig. 1 nd, respectively. In ripple-effect free logicl topology, there exists n exclusive (i.e., not shred) NIC in t lest one end of ech logicl link. For ech logicl link l 2 L, ifit shres n NIC on node n (i.e., if there exists i 2 I n such tht l 2 L n;i nd jl n;i j > 1), then we define s l ¼ n. If link l is etween nodes n nd m nd it does not shre n NIC on nodes n nd m (i.e., if there exist i 2 I n nd j 2 I m such tht l 2 L n;i \ L m;j ; jl n;i j¼1, nd jl m;j j¼1), then we ritrrily choose either s l ¼ n or s l ¼ m. In our model, for ech link l 2 L, wireless node s l is in chrge of the chnnel ssignment. Recll from Section 2 tht ech logicl link uses n exclusive NIC in t lest one end. Whenever node s l ssigns new chnnel to link l, no further chnnel ssignment is required in ny other node. Thus, the wireless nodes cn independently ssign the frequency chnnels of their corresponding logicl links. For the smple MC WMN topology in Fig. 1, we hve: s l1 ¼ s l2 ¼ s l3 ¼ s l4 ¼ nd s l5 ¼ s l6 ¼ s l7 ¼ s l8 ¼ c. On the other hnd, in Fig. 1, s l3 ¼ s l4 ¼ s l9 ¼ s l1 ¼ s l11 ¼ s l12 ¼ s l13 ¼ s l14 ¼ d. For ech node n 2 N, we define: x n ¼ðx l ; 8l 2 L; s l nþ: ð23þ Tht is, x n denotes the vector of chnnel ssignment vriles corresponding to ll logicl links other thn those links tht wireless node n is responsile for their chnnel ssignment. Given n ritrry chnnel ssignment vector ^x n, we lso define: n ð^x n Þ¼fx : x 2 ; x n ¼ ^x n g: ð24þ

8 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Tht is, n ð^x n Þ denotes the set of fesile chnnel ssignment vectors for ll logicl links l 2 L such tht s l ¼ n, ssuming tht fixed chnnels re ssigned to the rest of the logicl links ccording to ^x n. In our distriuted lgorithm, ech wireless node n, which is responsile to ssign the frequency chnnels to t lest one logicl link (i.e., 9 l 2 L such tht s l ¼ n), solves the following locl congestion-wre chnnel ssignment prolem: mximize x2nð^x nþ k l c l ðxþ; ðlocal-cacaþ l2l where the entries of ^x n re informed to node n y other nodes m 2 N nfng. Let L mx ¼ mx n2n jl n j. It is cler tht for ll n 2 N, prolem (LOCAL-CACA) is comintoril prolem with t most C Lmx comintions. As the network grows in size, L increses monotoniclly while L mx is lmost fixed. In prctice, L mx L. Thus, solving the locl prolem in (LOCAL-CACA) is significntly less complicted compred to solving the glol prolem in (CACA). Our proposed distriuted congestion-wre chnnel ssignment (DCACA) lgorithm is shown in Algorithm 1. For ech node n 2 N, we define T L;n T to denote the set of time slots t which prolem (LOCAL-CACA) is solved in node n. Any two consecutive memers of set T L;n re ssumed to e T L time units different. The wireless nodes solve prolem (LOCAL-CACA) synchronously. Tht is, for ll n; m 2 N, we hve T L;n \ T L;m ¼fg. In lines 2 5 of Algorithm 1, we initilize the lgorithm prmeters. In prticulr, we initilly set the network to operte on the first chnnel in single-chnnel scenrio. This is required to mke sure tht ll nodes cn primrily communicte with ech other to form the logicl topology. The logicl topology formtion in line 5 cn e performed similr to the steps explined in either [15] or [17]. Every T G time slots (see Section 2) nd in lines 7 nd 8, ech node mesures the congestion level t its outgoing links nd rodcsts the results to the rest of the network. By running lines 12 to 18, ech node independently solves prolem (LOCAL-CACA) using exhustive serch. After ssigning the optiml chnnels in line 2, node n informs the new chnnels to other nodes. We re now redy to show the following result: Theorem 1. If T L T G, Algorithm 1 converges to locl optimum of prolem (CACA). Proof. At ny time slot t 2 T, we define: wðtþ ¼ k l ðtþc l ðxðtþþ: l2l ð25þ Tht is, wðtþ denotes the ojective function of prolem (CACA) t time t. We first notice tht wðtþ is lwys ounded. In prticulr, we hve: wðtþ P ; 8t 2 T ð26þ nd wðtþ 6 1 log 1 þ Kgmx p mx ðlk T S g Þ; min 8t 2 T; ð27þ where g mx ¼ mx l2l g ll ; p l ¼ mx l2l p l ; g min ¼ min l2l g l, nd k mx ¼ mx l2l; t2t k l ðtþ. Note tht t ech time slot t 2 T; P l2l k lðtþ 6 ðlk mx Þ nd for ll logicl links l 2 L, we hve: c l ðxðtþþ 6 1 T S logð1 þðkg mx p mx Þ=g min Þ. We lso notice tht for ny t G 2 T G, the sclr function wðtþ is non-decresing during time slots ½t G þ 1; t G þ T G Š. To show this, consider n ritrry t 2½t G þ 1; t G þ T G Š. If t R [ n2n T L;n, then wðt þ 1Þ ¼wðtÞ. On the other hnd, if there exists wireless node n 2 N such tht t 2 T L;n, then xðt þ 1Þ is ssigned s the optiml solution of prolem (LO- CAL-CACA). Thus, xðt þ 1Þ is different from xðtþ only if wðt þ 1Þ is greter thn wðtþ. Otherwise, xðt þ 1Þ ¼xðtÞ nd wðt þ 1Þ ¼wðtÞ. Knowing tht wðtþ is ounded nd non-decresing, the convergence of Algorithm 1 is gurnteed s long s T G is lrge enough. Let x I 2 denote sttionry point of Algorithm 1. Also let w I 2½; 1=T S ðlk mx Þ logð1 þ Kg mx p mx =g min ÞŠ denote the vlue of the ojective function of prolem (CACA) t sttionry point x I. We first ssume tht x I is not locl optiml solution of prolem (CACA). Since the ojective functions in prolems (CACA) nd (LOCAL-CACA) re the sme, there should exist t lest one wireless node n 2 N such tht it cn prtilly devitex I to increse wðtþ; however, this contrdicts the fct tht x I is sttionry point. Thus, x I nd w I represent locl optiml solution nd locl optimum of prolem (CACA), respectively. h Algorithm 1 Distriuted congestion-wre chnnel ssignment (DCACA): To e executed y ech wireless mesh router n 2 N. 1: Allocte memory for x H ; w H ; k; q, nd ^x n. 2: Set k ¼ 1. 3: Set q ¼ 1. 4: Set w H ¼. 5: Set x H ¼ ½½1 Š½1 ŠŠ. 6: Form the logicl topology using the topology formtion lgorithms proposed in [15] or [17]. 7: for ll t 2 T do 8: if t 2 T G then 9: Set k n ¼ðk l ; 8 l 2 L out n Þ ccording to the congestion mesurements. 1: Set q n ¼ðq l ; 8 l 2 Ln out Þ ccording to the persistent proility mesurements. 11: Inform k n nd q n to ll nodes m 2 N nfng. 12: Set w H ¼. 13: end if 14: if t 2 T L;n then 15: for ll x 2 n ð^x n Þ do 16: Set c l ðxþ for ll l 2 L ccording to (22) given q. 17: Set w ¼ P l2l k lc l ðxþ. 18: if w > w H then 19: Set x H ¼ x. 2: Set w H ¼ w. 21: end if 22: end for 23: Assign the frequency chnnels ccording to x H. 24: Inform ^x n ¼ðx H l ; 8 l 2 L n ; s l ¼ nþ to ll nodes m 2 N nfng. 25: end if 26: end for

9 251 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) From Theorem 1, if prolem (CACA) only hs unique locl optiml solution, then Algorithm 1 will indeed converge to the est possile frequency chnnel ssignment ccording to prolem (CACA). In mny other cses, lthough there re more thn one locl optiml solutions, they re eqully good. For exmple, ssume tht C ¼ 2 nd L ¼ 3. In this cse, x ¼ ½½1 Š½ 1 Š½ 1ŠŠ T nd x ¼ ½½ 1Š½ 1Š½ 1 ŠŠ T result in the sme performnce. In fct, in oth cses, the first nd the second logicl links operte over the sme frequency chnnel while the third link opertes on different chnnel. We will further investigte the optimlity of Algorithm 1 nd its due effect on network performnce in Section 5. We lso notice tht sed on Algorithm 1, the wireless nodes re not selfish. In fct, they cooperte with ech other. This is indeed necessry for chieving the optiml network performnce in distriuted fshion. Assuming the cse where ech node n 2 N cts selfishly, it would solve the following selfish locl prolem: mximize x2nð^x nþ k l c l ðxþ; ðselfish-cacaþ l2l n where only the links of node n (i.e., the links in set L n ) re tken into ccount. By solving prolem (SELFISH-CACA), node n would not tke into ccount the interference tht its trnsmissions cuse on the trnsmissions from other nodes. This selfish ehvior hs een seen in vrious proposed chnnel ssignment strtegies in the literture. For exmple, ccording to the Lod-Awre chnnel ssignment strtegy in [15], ech node ssigns its links with the frequency chnnels which re less used y its neighoring trnsmissions, i.e., the est ville chnnels. However, in our proposed strtegy, some nodes my reserve chnnel for highly congested logicl link to help it to resolve its congestion prolem. As we will show in Section 5, the coopertion in Algorithm 1 mkes it noticely superior compred to the Lod-Awre lgorithm. 5. Performnce evlution In this section, we ssess the performnce of our proposed DCACA lgorithm sed on ns-2 simultions [45]. To support multiple NICs on ech wireless node, the ns-2 ptch from [46] is eing used. We modified the ptch so tht it cn lso support the prtilly-overlpped frequency chnnels. In the simultion model, the size of the network field is 1 m 1 m. Unless we specify otherwise, the MC WMN consists of 6 wireless mesh routers (i.e., N ¼ 6). Four of them serve s the gtewys, nd they re locted t the four corners in the field. Ten different topologies re generted. Topology numers 1; 2;...; 5 correspond to five different grid topologies, wheres topology numers 6; 7;...; 1 correspond to five different rndom topologies. For the grid topologies, the size of ech grid is 8 8. The distnce etween two djcent grid points is 14 m. Nodes re plced in 6 (out of 64) grid points rndomly. For the rndom topologies, nodes re rndomly plced in the network field such tht ech node hs t lest one neighor within its communiction rnge. Once the physicl topology hs een creted, the logicl topology is formed sed on the Hycinth ripple-effect free WMN rchitecture [15]. We lso set T G ¼ 6 s nd T L;n ¼ 5 s for ll n 2 N. In our simultion model, the trffic sources re ssumed to e TCP Vegs nd TCP Reno. For the cse when TCP Vegs sources re used, for ech logicl link l 2 L, the congestion mesure k l is its queuing dely. On the other hnd, for the cse when TCP Reno sources re used, for ech logicl link l 2 L, the congestion mesure k l is the link s pcket loss rte (i.e., the proility of dropping pcket). We modified the IEEE module in ns-2 so tht the higher lyer pplictions cn ccess the vector of the queueing delys nd pcket loss rte k. For given topology, in ech simultion run, 3 wireless nodes re rndomly selected s either the source (or destintion) for TCP flows to (or from) the Internet (i.e., the corresponding gtewy). The simultion time is 3 s. For ech wireless node, the chosen gtewy is the one which corresponds to the minimum hop pth. In our performnce evlution, we consider the following performnce metrics: (1) Aggregte throughput: totl numer of correctly received TCP segments (in its) t the destintions divided y the totl simultion time. (2) Averge round-trip time: verge time dely etween sending TCP segment nd receiving its cknowledgement. For TCP Vegs sources, we lso consider the ggregte network utility s the criterion to evlute the optimlity of our proposed lgorithm in terms of solving prolem (CACA), where k l denotes the queueing dely for ech link l 2 L. Notice tht TCP Vegs sources hve logrithmic utility functions s shown in [47]. For the cse of TCP Reno, lthough this protocol hs een reverse engineered (cf. [3]), finding n ccurte utility function is difficult tsk. Therefore, we evlute the optimlity of DCACA lgorithm only for the cse when TCP Vegs sources re eing simulted. The prmeters tht we used in the simultions re shown in Tle 2. Note tht most of them re the defult ns-2 prmeters. For the CSMA/CA (Crrier Sense Multiple Access) MAC protocol, the RTS/CTS (Request-To-Send/ Cler-To-Send) mechnism is enled. We lso considered the cse where ll the competing links in ech neighorhood re ssumed to hve n equl chnce to ccess the shred medium. Tht is, q l ¼ 1=ð1 þ O l Þ for ll links l 2 L. Tle 2 List of ns-2 simultion prmeters Trnsmission power ðp l Þ.2818 W Communiction rnge 25 m Crrier sensing rnge 45 m Receive threshold 3: W Crrier sensing threshold is 1: W Cpture threshold SINR min 1. Pth loss prmeter ðjþ 9: Therml noise power ðn l Þ 1: 1 11 W IEEE dt rte 54 Mps IEEE dt rte 11 Mps Queue type Drop-Til Queue size 5 Pkts TCP pcket size 1 Bytes TCP Vegs lph prmeter 1 TCP Vegs et prmeter 3 TCP Reno slow-strt threshold 2

10 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Comprison with Lod-Awre lgorithm We first compre the performnce etween DCACA nd Lod-Awre [15] lgorithms. Both lgorithms re distriuted nd run on top of the Hycinth ripple-effect free MC WMN logicl topologies. This leds to n ccurte nd fir comprison. We ssume tht there re two IEEE NICs in ech wireless mesh router (i.e., I ¼ I n ¼ 2 for ll n 2 N). We first limit our study to the cse where ll ville frequency chnnels re non-overlpping nd only six chnnels re ville (i.e., C ¼ 6). We will lter consider the cse where more chnnels nd NICs re ville nd lso the cse where the frequency chnnels re not only orthogonl, ut lso prtilly-overlpped in Section 5.2 nd 5.3, respectively. Fig. 4 shows the ggregte throughput nd the verge round-trip time for ll ten different topologies when TCP Vegs sources re eing used. Results from Fig. 4 show tht on verge, DCACA cn increse the ggregte throughput y 191.8% compred to the single-chnnel cse, nd y 11.5% compred to the multi-chnnel cse where the Lod-Awre lgorithm is used. Results from Fig. 4 lso show tht DCACA cn reduce the verge round-trip time y 234.1% compred to the single-chnnel cse, nd y 35.3% compred to the multi-chnnel cse where the Lod-Awre lgorithm is used. The superiority of DCACA, especilly on reducing the round-trip time, is evident. The etter performnce of DCACA Algorithm cn e explined sed on its fetures. Unlike the Lod-Awre lgorithm, where ech node selfishly tries to only improve its own performnce, DCACA lgorithm leds to glol coopertion mong the nodes (see Section 4). On the other hnd, our proposed lgorithm uses n ccurte link cpcity model which tkes into ccount vrious network prmeters such s trnsmission power, wireless pth loss, medium ccess control, nd the frequency response of the chnnel nd-pss filters. Next, we simulte the cse with presence of TCP Reno trffic. Results on the ggregte network throughput nd the verge round-trip time when TCP Reno sources re eing used re shown in Fig. 5. We cn see tht regrdless of the choice of the TCP protocol, our proposed DCACA lgorithm cn mnge to control congestion. Notice tht y chnging the TCP protocol, DCACA should chnge the choice of congestion price. In prticulr, for the cse of TCP Reno trffic, DCACA should consider the pcket loss rte s the congestion price on ech link. Results from Fig. 5 show tht on verge, DCACA cn increse the ggregte throughput y 177.2% compred to the single chnnel cse, nd y 9.8% compred to the multi-chnnel cse where the Lod-Awre lgorithm is used. Results from Fig. 5 lso show tht DCACA cn reduce the verge round-trip time y 181.3% compred to the single-chnnel cse, nd y 28.7% compred to the multi-chnnel cse where the Lod-Awre lgorithm is used. From [29], TCP Vegs ims to control the queueing dely long the routing pth of ech TCP session. In the next experiment, we vry the numer of TCP flows nd investi- 1 DCACA Lod Awre Single Chnnel Agg. Throughput (Mps) Avg. Round Trip Time (ms) DCACA Lod Awre Single Chnnel Agg. Throughput (Mps) Avg. Round Trip Time (ms) Fig. 4. Performnce comprison etween DCACA nd Lod-Awre distriuted chnnel ssignment lgorithms in presence of TCP Vegs trffic. () Aggregte throughput, () Averge round-trip time. Fig. 5. Performnce comprison etween DCACA nd Lod-Awre distriuted chnnel ssignment lgorithms in presence of TCP Reno trffic. () Aggregte throughput, () Averge round-trip time.

11 2512 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) gte the impct of numer of flows on the round-trip time. Fig. 6 nd show the verge round-trip time when the grid topology (with topology numer 1) nd the rndom topology (with topology numer 6) re eing simulted, respectively. Simultion results show tht y using single frequency chnnel, the network ecomes highly congested when the numer of TCP flows is greter thn ten. On the other hnd, y using two NICs nd ssigning six orthogonl frequency chnnels, the congestion is effectively voided in ll cses s long s our proposed DCACA is eing used. Note tht the DCACA lgorithm ssigns the pproprite chnnels in the neighorhood to increse the effective cpcities on the congested links. It thus prevents ny of the logicl links to ecome severely congested voiding lrge queuing delys Impct on ville resources There re two importnt resources in n MC WMN: the NICs t ech wireless mesh router, nd the ville frequency chnnels. To evlute the impct of the network resources, we vry the numer of NICs t ech router from 2 to 4, nd the numer of non-overlpped chnnels from 1 to 12 (IEEE frequency nd). Fig. 7 shows the ggregte throughput nd the verge round-trip time for the first rndom topology when TCP Vegs sources re eing used. The results for other rndom nd grid topologies re similr nd omitted for revity. We see tht the network performnce significntly increses s more resources re eing used. The improvements cn e interpreted in terms of the cpcity models in Section 3. For exmple, incresing the numer of NICs reduces the numer of logicl links tht shre common interfce. It removes some of the equlity constrints in (2) which consequently expnds the fesile set. On the other hnd, incresing the numer of frequency chnnels llows us to ssign different chnnels to ner-y links nd void mutul interference mong them. Thus, we cn hve more links l; k 2 L with x T l x k ¼. Compring the single-chnnel scenrio with multichnnel cse where C ¼ 12 nd I ¼ I n ¼ 4 for ll n 2 N, the DCACA lgorithm cn increse the ggregte throughput y fctor of 5.4 nd decrese the verge round-trip time y fctor of 5.6. Next, we study the impct of resources when TCP Reno sources re eing used. Fig. 8 shows the ggregte throughput nd the verge round-trip time in this cse. Agin, we cn see tht the network performnce increses s more resources re eing used. We cn conclude tht regrdless of the choice of TCP protocol, our proposed DCA- CA lgorithm cn properly utilize the ville network resources Performnce gin y using prtilly-overlpped chnnels There re 12 non-overlpped chnnels ville in the IEEE frequency nd which cn significntly increse the performnce, s discussed in Sections 5.1 nd 5.2. However, there re only 3 non-overlpped chnnels ville in the IEEE frequency nd. A smll numer of non-overlpped frequency chnnels cn limit the enefits of deploying n MC WMN. The performnce Avg. Round Trip Time(ms) Avg. Round Trip Time(ms) Single Chnnel Lod Awre DCACA Numer of TCP Flows Numer of TCP Flows Fig. 6. Averge round-trip time versus numer of estlished TCP Vegs flows. () Results for topology numer 1 (grid topology), () Results for topology numer 6 (rndom topology). Agg. Throughput (Mps) Avg. Round Trip Time (ms) Numer of Chnnels I = 4 I = 3 I = Numer of Chnnels Fig. 7. The performnce gin of incresing the numer of NICs per node nd the numer of ville orthogonl chnnels in the first rndom topology in presence of TCP Vegs trffic. () Aggregte network throughput, () Averge pcket round-trip time.

12 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Agg. Throughput (Mps) I = 4 I = 3 I = 2 Agg. Throughput (Mps) Using ll chnnels Using orthogonl chnnels Avg. Round Trip Time (ms) Numer of Orthogonl Chnnels Numer of Orthogonl Chnnels Fig. 8. The performnce gin of incresing the numer of NICs per node nd the numer of ville orthogonl chnnels in the first rndom topology in presence of TCP Reno trffic. () Aggregte network throughput, () Averge pcket round-trip time. cn e incresed y using ll the prtilly-overlpped chnnels. In this section, we evlute the performnce gin y using ll 11 ville chnnels in comprison with using only three orthogonl chnnels in the IEEE frequency nd. We consider the cse where there re two NICs in ech wireless mesh router. We use the rised cosine filter to model chnnel nd-pss filters nd set d ¼ 1 (see Fig. 2). The ggregte throughput nd the verge round-trip time when TCP Vegs source re used re shown in Fig. 9 nd, respectively. Note tht ecuse of the lower dt rte in IEEE compred to IEEE stndrd, the round-trip times in Fig. 9 re higher thn those in the previous figures. We see tht using the prtilly-overlpped chnnels is eneficil s it increses the ggregte network throughput y 25% nd decreses the verge round-trip time y more thn one hlf. The chieved performnce gin is due to n efficient usge of the frequency spectrum. Intuitively, DCACA ssigns two non-overlpped chnnels to the ner-y congested links, s long s the non-overlpped chnnels hve not een ssigned in the neighorhood. Otherwise, it ssigns two ville prtilly-overlpped chnnels tht cn cuse the minimum interference. Avg. Round Trip Time (ms) Next, we study the performnce gin of using prtillyoverlpped chnnels in presence of TCP Reno trffic. Results re shown in Fig. 1. We cn see tht regrdless of the choice of TCP protocol, using ll ville prtillyoverlpped chnnels cn improve the performnce compred to the cse when only the orthogonl chnnels re eing used. Note tht ecuse of the lower dt rte in IEEE compred to IEEE stndrd, the roundtrip times in Fig. 9 re higher thn those in the previous figures. We see tht using the prtilly-overlpped chnnels cn increse the ggregte network throughput y 27.6% nd decreses the verge round-trip time y 78.5% Optimlity Fig. 9. Performnce gin y using ll eleven prtilly-overlpped chnnels in IEEE frequency nd insted of only using the three nonoverlpped chnnels 1, 6, nd 11 in presence of TCP Reno trffic. There re two interfces in ech wireless router ði ¼ 2Þ. () Aggregte throughput, () Averge round-trip time. Recll from Section 2 tht solving prolem (CACA) helps to solve other resource lloction prolems such s network utility mximiztion nd mximum weight mtching. The former is n importnt design ojective in the presence of elstic trffic sources. In this section, we evlute the cpility of DCACA in solving the network utility mximiztion prolem cross TCP Vegs sources. Unlike TCP Reno sources, the TCP Vegs sources re designed to mximize specific utility function which is logrithmic [47]. We consider the ggregte network utility s the performnce metric in this section. Assume tht there exists TCP Vegs source from node n 2 N to node d 2 N nfng. Let r sd denote its trnsmission rte. The utility of this TCP Vegs source is then defined s D sd logðr sd Þ, where D sd de-

13 2514 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) Agg. Throughput (Mps) Avg. Round Trip Time (ms) Using ll chnnels Using orthogonl chnnels Fig. 1. Performnce gin y using ll eleven prtilly-overlpped chnnels in IEEE frequency nd insted of only using the three non-overlpped chnnels 1, 6, nd 11 in presence of TCP Reno trffic. There re two interfces in ech wireless router ði ¼ 2Þ. () Aggregte throughput, () Averge round-trip time. notes the fixed dely t the routing pth from node s to node d [47]. The ggregte network utility is then defined s P P s2n d2nnfsg D sd logðr sd Þ. To otin the optiml utility for ech topology, we simulte ll the fesile chnnel ssignments nd select the mximum mesured network utility. We consider five rndom nd five grid topologies. Ech topology includes 15 nodes (i.e., N ¼ 15) nd hs one gtewy. Notice tht there is no limittion on running DCACA for lrge-scle MC WMNs s we showed in the previous experiments. However, to e le to otin the exct optiml network utility, we need to exmine ll the chnnel ssignment possiilities nd there re C L comintions. This required us to limit the network size. Nevertheless, our study here provides enchmrk to evlute the optimlity of DCACA lgorithm. In our simultion model, ten nodes re rndomly selected s TCP sources. Ech router is equipped with two IEEE NICs. We use three orthogonl chnnels. The rest of the simultion settings re the sme s efore. Results from Fig. 11 show tht our proposed DCACA lgorithm cn led to 99.4% optimlity on verge. To hve etter understnding on the effect of the optimlity gps on network performnce, we hve lso shown the ggregte network throughput nd the verge round-trip time in Fig. 11 nd c, respectively. On verge, using Algorithm Network Utility c Agg. Throughput (Mps) Avg. Round Trip Time (ms) Optiml DCACA Fig. 11. Optimlity of DCACA to solve the network utility mximiztion prolem in presence of TCP Vegs trffic sources where for ech link, the corresponding congestion mesure is indeed the link s queueing dely. () Network Utility, () Aggregte throughput, (c) Averge round-trip time. We see tht DCACA Algorithm results in ner optiml network utilities in ll cses. 1, the performnce degrdtion on ggregte network throughput nd verge round-trip time re only 6.3% nd 7.9%, respectively. Thus, our proposed distriuted congestion-wre chnnel ssignment lgorithm cn led to ner optiml solution for the network utility mximiztion prolem. On the other hnd, s shown in Section 5.1, our proposed lgorithm cn significntly improve the network performnce compred to the Lod-Awre distriuted chnnel ssignment strtegy. In prticulr, the performnce further improves if we use not only the non-overlpped chnnels, ut lso the ll ville prtilly-overlpped chnnels. 6. Conclusions nd future work In this pper, we considered the prolem of mximizing weighted summtion of ll link cpcities where for ech link, the corresponding weighting prmeter is the link s congestion mesure. Vrious congestion mesures cn e considered such s queueing dely, pcket loss rte, or differentil cklog. We first otined comprehensive

14 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) closed-form mthemticl link cpcity model in terms of our defined chnnel ssignment vriles. We lso introduced the chnnel overlpping nd mutul interference mtrices to model the effect of prtil frequency overlpping mong the chnnels. Unlike most of the previous chnnel ssignment lgorithms, our proposed scheme ssigns not only the orthogonl (i.e., non-overlpped) frequency chnnels, ut lso the prtilly-overlpped chnnels. We then proposed distriuted congestionwre chnnel ssignment lgorithm (DCACA) which works synchronously mong the wireless mesh routers nd requires ech node only to execute simple locl serch procedure. To ssess the performnce of DCACA lgorithm, we performed extensive ns-2 network simultions for oth grid nd rndom MC WMN topologies. In the presence of TCP Vegs trffic sources nd when the congestion mesure of ech link is selected to e the corresponding link s queueing dely, our proposed lgorithm increses the ggregte throughput y 11.5% nd decreses the verge pcket round-trip time y 35.3% compred to the Lod-Awre chnnel ssignment lgorithm [15]. In the presence of TCP Reno trffic sources nd when the congestion mesure of ech link is selected to e the corresponding link s pcket loss rte, our proposed lgorithm increses the ggregte throughput y 9.8% nd decreses the verge pcket round-trip time y 28.7% compred to the Lod-Awre chnnel ssignment lgorithm. In congested IEEE network setting, compred with the cse where we only used the three non-overlpped chnnels, the ggregte network throughput cn further e incresed y 25% nd the verge round-trip time cn e further reduced y more thn one hlf when ll the 11 prtilly-overlpped chnnels re used. The current work cn e extended in severl directions. In prticulr, we cn further consider the impct of using directed ntenn to further reduce the interference. In this regrd, we cn use the interference models in [23] nd reformulte the congestion-wre chnnel ssignment prolem in (CACA) ccordingly. We shll lso evlute our proposed DCACA lgorithm through test-ed study. In prticulr, it is importnt to ssess the lgorithm performnce in presence of comintion of oth TCP nd UDP trffic. Finlly, the proposed joint congestion-wre chnnel ssignment scheme cn e further improved y dding power control. Notice tht the interference cn e reduced not only y ssigning distinct chnnels to neighoring trnsmissions ut lso properly djusting the trnsmission power of ech node. References [1] I. Akyildiz,. Wng, A survey on wireless mesh networks, IEEE Communictions Mgzine 43 (Septemer) (25) [2] R. Bruno, M. Conti, E. Gregori, Mesh networks: commodity multihop d hoc networks, IEEE Communictions Mgzine 43 (Mrch) (25) [3] P. Bhl, A. Ady, J. Pdhye, A. Wolmn, Reconsidering wireless systems with multiple rdios, ACM Computer Communictions Review 34 (Octoer) (24) [4] H.S. Chiu, K.L. Yeung, K.S. Lui, On optimiztion of joint chnnel ssignment nd routing in moile d hoc networks, in: Proceedings of IEEE Gloecom, Wshington, DC, 27. [5] A.K. Ds, H.M.K. Alzemi, R. Vijykumr, S. Roy, Optimiztion models for fixed chnnel ssignment in wireless mesh networks with multiple rdios, in: Proceedings of IEEE SECON, Snt Clr, CA, 25. [6] Y.Y. Chen, S.C. Liu, C. Chen, Chnnel ssignment nd routing for multi-chnnel wireless mesh networks using simulted nneling, in: Proceedings of IEEE Gloecom, Sn Frncisco, CA, 26. [7] A.H. Mohsenin Rd, V.W.S. Wong, Joint logicl topology design, interfce ssignment, chnnel lloction, nd routing for multichnnel wireless mesh networks, IEEE Trnsctions on Wireless Communictions 6 (Decemer) (27) [8] Y.H. Tm, R. Benkoczi, H.S. Hssnein, S.G. Akl, Optiml chnnel ssignment in multi-hop cellulr networks, in: Proceedings of IEEE Gloecom, Wshington, DC, 27. [9] H. Zhou, C. Yeh, H.T. Moufth, A relile low-overhed mc protocol for multi-chnnel wireless mesh networks, in: Proceedings of IEEE Gloecom, Wshington, DC, 27. [1] N.H. Trn, C.S. Hong, Joint scheduling nd chnnel lloction in wireless mesh networks, in: Proceedings of IEEE CCNC, Ls Vegs, NV, 28. [11] W. Li, P. Fn, K.B. Letief, A comintion scheme of topology control nd chnnel ssignment in wireless d hoc networks, in: Proceedings of IEEE Gloecom, Wshington, DC, 27. [12] M. Alicherry, R. Bhti, L.E. Li, Joint chnnel ssignment nd routing for throughput optimiztion in multi-rdio wireless mesh networks, IEEE Journl on Selected Ares in Communictions 24 (Novemer) (26) [13] M. Kodilm, T. Nndgopl, Chrcterizing the cpcity region in multi-rdio multi-chnnel wireless mesh networks, in: Proceedings of ACM MoiCom, Cologne, Germny, 25. [14] Y. Song, C. Zhng, Y. Fng, Throughput mximiztion in multichnnel wireless mesh ccess networks, in: Proceedings of IEEE ICNP, Beijing, Chin, 27. [15] A. Rniwl, T. Chiueh, Architecture nd lgorithms for n IEEE sed multi-chnnel wireless mesh network, in: Proceedings of IEEE Infocom, Mimi, Florid, 25. [16] A. Rniwl, K. Gopln, T. Chiueh, Centrlized chnnel ssignment nd routing lgorithms for multi-chnnel wireless mesh networks, ACM Moile Computing nd Communiction Review 8 (April) (24) [17] A.H. Mohsenin Rd, V.W.S. Wong, Logicl topology design nd interfce ssignment for multi-chnnel wireless mesh networks, in: Proceedings of IEEE Gloecom, Sn Frncisco, CA, 26. [18] G. Zeng, B. Wng, Y. Ding, L. io, M. Mutk, Multicst lgorithms for multi-chnnel wireless mesh networks, in: Proceedings of IEEE ICNP, Beijing, Chin, 27. [19] A. Ady, P. Bhl, J. Pdhye, A. Wolmn, L. Zhou, A multi-rdio unifiction protocol for IEEE wireless networks, in: Proceedings of IEEE Brodnet 4, Sn Jose, CA, 24. [2] S. Shrm, N. Zhu, T. Chiueh, Low-ltency moile IP hndoff for infrstructure-mode wireless LANs, IEEE Journl on Selected Ares in Communictions 22 (My) (24) [21] A. Mishr, V. Shrivstv, S. Bnerjee, W. Arugh, Prtilly overlpped chnnels not considered hrmful, in: Proceedings of ACM SIGMetric, Sint Mlo, Frnce, 26. [22] A.H. Mohsenin Rd, V.W.S. Wong, Joint optiml chnnel ssignment nd congestion control in multi-rdio wireless mesh networks, in: Proceedings of IEEE ICC, Istnul, Turkey, 26. [23] B. Gvish, Y. Ofek, R. Whitker, Uplink nlysis with moile devices using directionl-smrt ntenns, Interntionl Journl of Moile Network Design nd Innovtion 1 (25) [24] M. Felegyhzi, M. Cglj, S.S. Bidokhti, J.P. Huux, Non-coopertive multi-rdio chnnel lloction in wireless networks, in: Proceedings of IEEE INFOCOM, Anchorge, Alsk, 27. [25] F. Kelly, Chrging nd rte control for elstic trffic, Europen Trnsctions on Telecommuniction 8 (1997) [26] M. Ching, S.H. Low, A.R. Cldernk, J.C. Doyle, Lyering s optimiztion decomposition: A mthemticl theory of network rchitectures, in: Proceedings of IEEE, vol. 95, 27, pp [27] L. Chen, S.H. Low, J. Doyle, Joint congestion control nd medi ccess control design for d hoc wireless networks, in: Proceedings of IEEE Infocom, Mimi, Florid, 25. [28] J. Wng, L. Li, S.H. Low, J. Doyle, Cross-lyer optimiztion in TCP/IP networks, IEEE/ACM Trnsctions on Networking 13 (June) (25) [29] L.S. Brkmo, L.L. Peterson, TCP Vegs: end to end congestion voidnce on glol internet, IEEE Journl on Selected Ares in Communictions 13 (Octoer) (1995)

15 2516 A.H. Mohsenin Rd, V.W.S. Wong / Computer Networks 53 (29) [3] S. Low, F. Pgnini, J. Doyle, Internet congestion control, IEEE Control Systems Mgzine (Ferury) (22) [31] L. Tssiuls, A. Ephremides, Stility properties of constrined queueing systems nd scheduling policies for mximum throughput in multihop rdio networks, IEEE Trnsctions on Automtic Control 13 (June) (1992) [32] M.J. Neely, E. Modino, C.E. Rohrs, Dynmic power lloction nd routing for time-vrying wireless networks, IEEE Journl on Selected Ares in Communictions 23 (Jnury) (25) [33] J. Prokis, Dt Communictions, fourth ed., McGrw-Hill, 2. [34] D. Tse, P. Viswnth, Fundmentls of Wireless Communiction, Cmridge University Press, 24. [35] M. Ching, To lyer or not to lyer: lncing trnsport nd physicl lyers in wireless multihop networks, in: Proceedings of IEEE Infocom, Hong Kong, Chin, 24. [36] Y. i, E.M. Yeh, Throughput optiml distriuted control of stochstic wireless networks, in: Proceedings of the Fourth Interntionl Symposium on Modeling nd Optimiztion in Moile, Ad Hoc nd Wireless Networks (WiOpt 6), Boston, MA, 26. [37] K. Krkyli, J. Kng, M. Kodilm, K. Blchndrn, Joint resource lloction nd routing for ofdm-sed rodnd wireless mesh networks, in: Proceedings of IEEE ICC, Glsgow, Scotlnd, 27. [38] J. Lee, M. Ching, R. Cldernk, Utility-optiml rndom-ccess control, IEEE Trnsctions on Wireless Communictions 25 (August) (27) [39] A.H. Mohsenin Rd, J. Hung, M. Ching, V.W.S. Wong, Utilityoptiml rndom ccess: Reduced complexity, fst convergence, nd roust performnce, IEEE Trnsctions on Wireless Communictions 8 (Ferury) (29) [4] Y.C. Ty, K.C. Chu, A cpcity nlysis for the IEEE MAC protocol, Wireless Networks 7 (Mrch) (21) [41] J. Lee, A. Tng, J. Hung, M. Ching, A. Cldernk, Reverse engineering MAC: gme-theoretic model, IEEE Journl on Selected Ares in Communictions 6 (July) (27) [42] D.P. Bertseks, R. Gllger, Dt Communictions, second ed., Prentice Hll, [43] R. Horn, C. Johnson, Topics in Mtrix Anlysis, first ed., Cmridge University Press, [44] H. Th, Opertions Reserch: An introduction, seventh ed., Prentice Hll, 23. [45] The ns2 network simultor. < [46] ns2 ptch. < multi-nic-ptch>, relesed 24. [47] S.H. Low, L. Peterson, L. Wng, Understnding vegs: dulity model, Journl of the ACM 49 (Mrch) (22) Amir-Hmed Mohsenin-Rd received his B.Sc. degree from Amir-Kir University of Technology (Tehrn, Irn) in 22, M.Sc. degree from Shrif University of Technology (Tehrn, Irn) in 24, nd Ph.D. degree from the University of British Columi (Vncouver, Cnd) in 28, ll in electricl engineering. From Mrch to July 27, he ws lso visiting scholr t Princeton University (Princeton, NJ). As grdute student, he received the UBC Grdute Fellowship s well s the Pcific Century Grdute Scholrship from the British Columi Provincil Government. He currently serves s TPC memer for the IEEE Interntionl Conference on Communictions (ICC9) nd the IEEE Consumer Communictions nd Networking Conference (CCNC9). His reserch interests re in the re of optimiztion theory nd its pplictions in computer communictions nd wireless networking. Vincent W.S. Wong received the B.Sc. degree from the University of Mnito, Winnipeg, MB, Cnd, in 1994, the M.A.Sc. degree from the University of Wterloo, Wterloo, ON, Cnd, in 1996, nd the Ph.D. degree from the University of British Columi (UBC), Vncouver, BC, Cnd, in 2. From 2 to 21, he worked s systems engineer t PMC-Sierr Inc. He joined the Deprtment of Electricl nd Computer Engineering t UBC in 22 nd is currently n Associte Professor. His reserch interests re in resource nd moility mngement for wireless mesh networks, wireless sensor networks, nd heterogeneous wireless networks. He is n ssocite editor of the IEEE Trnsctions on Vehiculr Technology nd n editor of KICS/IEEE Journl of Communictions nd Networks. He serves s TPC memer in vrious conferences, including IEEE Gloecom, ICC, nd Infocom. He is senior memer of the IEEE nd memer of the ACM.

P(r)dr = probability of generating a random number in the interval dr near r. For this probability idea to make sense we must have

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