On channel assignment and multicast routing in multi-channel multi-radio wireless mesh networks. Mohsen Jahanshahi*

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1 Int. J. Ad Hoc and Ubqutous Computng, Vol. 12, No. 4, On channel assgnment and multcast routng n mult-channel mult-rado wreless mesh networks Mohsen Jahanshah* Computer Engneerng Department, Scences & Research Branch, Islamc Azad Unversty, P.O. Box: , Hesarak, Tehran, I.R. Iran, E-mal: mjahanshah@auctb.ac.r *Correspondng author Mehd Dehghan and Mohammad Reza Meybod Computer Engneerng Department, Amrkabr Unversty of Technology, No. 424, Hafez Ave, P.O. Box: , Tehran, Iran E-mal: dehghan@aut.ac.r E-mal: mmeybod@aut.ac.r Abstract: Multcast s a key networkng servce, enablng one-shot delvery of nformaton from a source to multple destnatons and s consdered underlyng for collaboratng multmeda applcatons such as vdeo conferencng, dstance learnng and other forms of content dstrbutng over Mult-Channel Mult-Rado Wreless Mesh Networks (MCMR WMNs. Multcast protocol as desgned n these networks, however, s tghtly coupled wth the specfcs of the nodes channel-rado assocatons to realse mnmum nterference communcaton. The manstream of research n WMN multcastng s orented towards heurstc or meta-heurstc strateges whch bascally take on a sequental approach to solve the channel assgnment and the multcast routng as two dsjont sub-problems. The resultng network confguratons would be sub-optmal n ths case. It s gven that the cross-nteracton between the two sub-problems s an effect of the problem s specfcatons. In ths paper, frst, we propose a cross-layer mathematcal formulaton of jont channel assgnment and multcast tree constructon n MCMR WMNs, whch, opposed to the exstng schemes guarantees optmal soluton. The smulaton results demonstrate that our cross-layer desgn outperforms the Level Channel Assgnment (LCA, Mult-Channel Multcast (MCM, the Genetc Algorthm (GA, Smulated Annealng (SA and the Tabu Search (TS-based methods proposed by Zeng et al. (2010 and Cheng et al. (2011 respectvely, n terms of nter-channel nterference. Second, snce jont optmsaton modellng has been relatvely demandng n terms of complexty, we relax the optmalty requrement and alternatvely explore the opton of a layered formulaton n whch to ensure an optmal soluton for each sub-problem. Our alternatve desgn s proved superor to the pror art n terms of nterference mnmsaton too. We conduct an extensve seres of smulatons to analyse the optmalty and complexty of our two desgn strateges. The overall result of the nterference, s our optmalty measurement. Also, complexty s evaluated n terms of the memory consumpton as well as the requred tme to solve the multcast problem. Keyword: WMN; wreless mesh network; mult-channel; mult-rado; channel assgnment; multcast tree constructon; cross-layer desgn; BIP; bnary nteger programmng. Reference to ths paper should be made as follows: Jahanshah, M., Dehghan, M. and. Meybod, M.R. (2013 On channel assgnment and multcast routng n mult-channel multrado wreless mesh networks, Int. J. Ad Hoc and Ubqutous Computng, Vol. 12, No. 4, pp Bographcal notes: Mohsen Jahanshah completed hs BS and MS n Computer Engneerng n Iran n 2002 and 2005, respectvely. He also acheved a PhD n Computer Engneerng from the Islamc Azad Unversty (Tehran Scence and Research Branch n He joned the faculty of the Computer Engneerng Department at Islamc Azad Unversty (Central Tehran Branch n Hs research nterests nclude wreless mesh networks, wreless sensor networks, cogntve networks, mathematcal optmsaton, learnng systems and soft computng. Copyrght 2013 Inderscence Enterprses Ltd.

2 226 M. Jahanshah et al. Mehd Dehghan receved hs BS n Computer Engneerng from the Iran Unversty of Scence and Technology (IUST, Tehran, Iran, n 1992 and hs MS and PhD from Amrkabr Unversty of Technology (AUT, Tehran, Iran, n 1995 and 2001, respectvely. He s an Assocate Professor of Computer Engneerng and Informaton Technology at Amrkabr Unversty of Technology (AUT. Pror to jonng AUT n 2004, he was a Research Scentst at the Iran Telecommuncaton Research Center (ITRC, workng n the area of qualty-of-servce provsonng and network management. Hs research nterests are n wreless networks, pattern recognton, fault-tolerant computng and dstrbuted systems. M.R. Meybod receved hs BS and MS n Economcs from the Shahd Behesht Unversty n Iran n 1973 and 1977, respectvely. He also receved hs MS and PhD n Computer Scence from Oklahoma Unversty, USA, n 1980 and 1983, respectvely. Currently, he s a Full Professor n the Computer Engneerng Department of Amrkabr Unversty of Technology, n Tehran, Iran. Pror to hs current poston, he worked from1983 to 1985 as Assstant Professor at Western Mchgan Unversty and from 1985 to 1991 as Assocate Professor at Oho Unversty, USA. Hs research nterests nclude channel management n cellular networks, learnng systems, parallel algorthms, soft computng and software development. 1 Introducton The Wreless Mesh Network (WMN has been envsoned as the economcally vable networkng paradgm for scalable QoS-aware delvery of heterogeneous traffc over broadband and large-scale wreless commodty networks (Wang et al., 2005; Martnez and Bafalluy, WMNs consst of mesh routers and mesh clents, where mesh routers, as opposed to nodes n a moble/statc ad hoc network (MANET (Jeong et al., 2007; Yang and Wang, 2009; Rabara and Vjayalekshm, 2011; Palansamy and Annadura, 2011; Cao et al., 2009; Mohamed and Alnuwer, 2009 have mnmal moblty and form the backbone of WMNs. They also provde network access for both mesh and conventonal clents (Wang et al., Compared to Wreless Sensor Networks (WSNs (Ye and Cheng, 2007, mesh nodes are not energy constraned. However, the man objectve n these networks s physcal layer capacty maxmsaton through the nterference mnmsaton, whch s typcally acheved by havng each node transceve through multple rados tuned on multple channels (Bahl et al., 2004; Ma et al., 2008 thus beng a partcpant n a number of parallel communcatons (Gupta and Kumar, Some of the key applcatons for Mult-Channel Mult-Rado WMNs (MCMR WMNs are multcastbased systems such as vdeo conferencng, onlne games, webcast and dstance learnng, etc. However, whle wreless communcaton s ntrnscally apt for performng multcast routng due to the broadcast nature of the ar medum, nter-channel nterference n WMNs plays a key factor n determnng the actual data rate achevable for a multcast servce. Wreless nterference occurs when two lnks wthn a two-hop dstance of each other are assgned to the same channel. The nterference caused by channel conflct dmnshes the performance of the wreless communcaton dramatcally. Therefore, for multcast routng, each lnk on the multcast tree should be assgned to a channel such that mnmal nterference occurs. As clearly stpulated n the Protocol model (Gupta and Kumar, 2000; Kodalam and Nandagopal, 2005; Hossan and Leung, 2008, whch s also the nterference model underlyng the formulaton presented n ths paper, mnmsng the number of nterferng lnks n a wreless mesh network would have a drect mpact on throughput maxmsaton, as also corroborated n Zeng et al. (2007, 2010 and Cheng and Yang (2011. Clearly, multcast routng n WMNs not only the best routng tree s establshment requres but we also need to assgn proper channels to ts lnks. In MCMR WMNs, nterference mnmsaton s normally tackled by devsng channel assgnment schemes n the MAC layer. Effcent channel-rado assocatons often come at the expense of satsfyng more complex connectvty constrants; n partcular, for two nodes to be connected, not only should they be wthn transmsson radus of one another, but also that the same channel should be assgned to ther rados. Multcast routng, under ths addtonal constrant, s even more complcated, snce the multcast tree now has to be constructed for the channel-rado assocaton, whch also mnmses nter-channel nterference. Despte ts vast number of applcatons and practcal mportance, few works have specfcally been targeted at multcast performance optmsaton n MCMR WMNs. The manstream of research n ths area has consdered channel assgnment and multcast routng as two dsjont sub-problems to be solved n sequence (Zeng et al., 2007, 2010; Cheng and Yang, 2008a, 2008b, 2011; Lm et al., 2009, as envsaged n Nguyen and Nguyen (2008, 2009a, 2009b and Yn et al. (2007, t mght even be the case that the soluton for ether one of these two sub-problems s assumed to be prevously calculated and gven as nput to the other. The downsde assocated wth these schemes, however, s that the cross-nteracton between the two sub-problems would not be accounted for and that ther relance on heurstc or meta-heurstc ntatves does not come up wth the optmal soluton. Networkng problems are deally formulated n mathematcal terms so that the resultng confguratons

3 On channel assgnment and multcast routng n mult-channel mult-rado wreless mesh networks 227 and performance tunng parameters can be optmsed wth ultmate guarantee. However, to date, we are unaware of any publshed study that has mathematcally approached the cross-optmsaton of channel assgnment and multcast tree constructon problems for MCMR WMNs. In ths paper, we propose a cross-layer desgn for jont optmsaton of channel assgnment and multcast tree constructon problems. As opposed to the pror art, the two sub-problems would be solved conjontly and ther nterplay s thoroughly accounted for. The mathematcal formulaton adopted n ths paper s based on a Bnary Integer Programmng (BIP model whch, compared to the prevous heurstc or meta-heurstc-based models, guarantees an optmal soluton. We argue that lnear programmng-based formulatons sut the typcally low-densty and lmted scale of WMN settngs (Nguyen and Xu, 2007; Nguyen, Such formulatons also come wth the advantage of easy extensblty n the sense that ther constrant-based problem defnton can smply be talored to ft a new desgn context. The applcaton of BIP, especally n our cross-optmsaton settng, yelds a channel assgnment scheme whch fully exhausts the pool of avalable resources (.e., channels and rados so as to construct the optmum (vz. mnmal nterference multcast routng structure. Our modellng essentally provdes a frame of reference to evaluate comparable centralsed or dstrbuted schemes by gvng the optmalty of the outcome of a BIP formulaton. Smulaton results reveal that our proposed crosslayer desgn outperforms the two heurstc-based schemes dscussed n Zeng et al. (2007, 2010, namely Level Channel Assgnment (LCA and Mult-Channel Multcast (MCM, as well as the methods based on the Genetc Algorthm (GA, Smulated Annealng (SA and the Tabu Search (TS proposed n Cheng and Yang (2008a, 2008b, 2011, n terms of nterference sufferng by the network. Furthermore, snce jont optmsaton modellng s relatvely demandng n terms of complexty, we relax the optmalty requrement and alternatvely explore the opton of a layered formulaton to ensure an optmal soluton for each sub-problem nstead. As evdenced by the outcome of the smulatons, the nterference assocated wth the layered approach, though margnally hgher than that of the cross-layer desgn, s stll sgnfcantly smaller compared to those of LCAs, MCMs and methods based on GA, SA and TS. The effcency of our framework s further nvestgated by contrastng the characterstcs of our layered and cross-layer desgns n terms of both optmalty and complexty. The metrcs used to measure the optmalty of the two proposed approaches ncludes the nterference suffered by the network. On the other hand, complexty s measured n terms of memory consumpton as well as the tme requred to solve the multcast problem. In addton, the tme complexty of two proposed methods s analysed. We also explctly address the hdden channel problem (Lm et al., 2009 whch occurs when two nodes wthn a twohop dstance, whch are tuned on the same channel, attempt to send or receve data smultaneously. The remnder of ths paper s organsed as follows: In Secton 2, we brefly revew the exstng multcast routng methods n MCMR WMNs, hghlghtng ther strengths and shortcomngs. Detals of the mathematcal formulatons for the proposed layered and cross-layered schemes are presented n Secton 3. In Secton 4, the ssues of connectvty and loop occurrence are nvestgated for our framework. Secton 5 s dedcated to a comparatve smulaton study together wth the analyss of the optmalty and complexty of our solutons. In Secton 6, two proposed desgns are further nvestgated wth respect to tme complexty and memory demands. We conclude the paper n Secton 7. 2 Related works As also ponted out n the Introducton, to acheve hgh performance n WMNs, nterference should be reduced. One of the most sgnfcant nterference reducton technques s channel assgnment, whch specfes the channel to be assgned to a rado n such a way that there s the least contrbuton to the overall nterference. Channel assgnment for uncast routng n MCMR WMNs has been addressed extensvely n the lterature (e.g., Kodalam and Nandagopal (2005, Skall et al. (2007, Subramanan et al. (2008, Rad and Wong (2006, Ramachandran et al. (2006, Marna et al. (2010, Das et al. (2005, Alcherry et al. (2005, Tang et al. (2005, Ranwala et al. (2004 and Tasak et al. (2004. The manstream of research n ths area can be classfed n two categores. The frst s dsjont, n whch channel assgnment s performed on a gven routng topology (Rad and Wong, 2006; Das et al., 2005; Ranwala et al., 2004, or routng s accomplshed over a gven channel assgnment scheme (Subramanan et al., 2008; Ramachandran et al., 2006; Marna et al., 2010; Tang et al., The second s jont methods (Kodalam and Nandagopal, 2005; Alcherry et al., Obvously, uncast-based mplementatons are not readly applcable, or at least scalable enough to be employed n the one-to-many paradgm of a typcal multcast communcatons settng. Moreover, gven the bandwdth-constraned operaton of wreless networks, exstng wrelne multcast solutons cannot be ported to mesh systems wthout fundamentally changng ther behavour to reduce overhead. Multcastng n MANETs and WSNs also addresses route recovery and energy concerns, whch are characterstcally dfferent from the pvotal ssues of throughput and nterference rased n the mddle-layer of MCMR WMNs. Routng n these networks s further complcated, gven that the multple rados on each node may dynamcally swtch on dfferent channels. WMNbased multcastng has been dscussed n Keegan et al. (2008, Ruz and Gomez-skarmeta (2005, Roy et al. (2008, Zhao et al. (2006, Yuan et al. (2006, Shttu et al. (2008, Ruz et al. (2006, Akyldz and Wang (2008 and Crchgno et al. (2008, albet for sngle-channel sngle-rado scenaros. The work n Karm et al. (2010, on the other hand, targets multchannel sngle-rado settngs, whch characterse sgnfcantly dfferent network confguratons; further, the major emphass n Karm et al. (2010 s placed on throughput maxmsaton, of whch multcast tree constructon s not a necessty,

4 228 M. Jahanshah et al. as opposed to the problem of nterest n ths paper, whch essentally narrows down to jont channel assgnment and multcast tree constructon wth mnmal nterference. In Gopnathan et al. (2009, multcast throughput optmsaton n MCMR WMNs s modelled n terms of an Integer Lnear Programmng (ILP formulaton. To acheve the optmal result, the nterplay between channel assgnment n the MAC layer and multcast routng n the network layer should be accounted (Akyldz and Wang, 2008; ths has been neglected n Gopnathan et al. (2009. Moreover, unlke the case n our desgn, a multcast tree constructon does not form the manstay of Gopnathan et al. (2009. In our bref revew of the relevant lterature, we specfcally focus on studes addressng both channel assgnment and multcast tree constructon n MCMR WMNs. Exstng schemes may be roughly classfed nto the followng three categores: methods takng channel assgnment for granted, thus treatng multcast tree constructon as the man problem (Nguyen and Nguyen, 2009; methods assumng a gven multcast tree, thus solvng for channel assgnment as the core problem (Nguyen and Nguyen, 2008a, 2009; Yn et al., 2007; and fnally, methods sequentally solvng for both multcast tree constructon and channel assgnment (Zeng et al., 2007, 2010; Cheng and Yang, 2008a, 2008b, 2011; Lm et al., In Zeng et al. (2007 and ts extended verson n Zeng et al. (2010, two methods for multcast tree constructon and channel assgnment n MCMR WMNs have been proposed. In the frst method, mesh nodes are ntally vsted by conductng a BFS startng from the multcast source. Ths way, nodes are placed at dfferent levels from source to multcast group members. Forwardng nodes n the multcast tree, on the other hand, are specfed by takng on a bottomup approach: f each recever node v has several parents and one of ther parents s on the multcast tree, ths recever s connected to that parent (f v. Otherwse, one of the parent nodes s selected randomly and one lnk s establshed to that parent (f v. The algorthm for node f v would contnue recursvely. After constructng the multcast tree, a so-called LCA algorthm s used, whch assgns channels to nodes dependng on what level of BFS traversal tree they belong to. In partcular, channel s assgned to nodes located at level of the tree. Whle the man advantage of ths scheme les n ts smplcty, t also comes wth the followng shortcomngs: when presented wth multple canddates, a multcast recever has to be content wth a parent chosen at random, whch may not always work out to be the most promsng choce. Also, there s no dscusson as to whch recever node should ntate tree constructon, despte the mplcatons t mght have on the specfcs of the resultant tree structure. As for channel assgnment, f the number of channels s more than the number of levels, the pool of avalable channels wll be left underutlsed. The second method, namely MCM also starts wth placng the tree nodes at dfferent levels usng BFS. The mnmum number of Relay Nodes (RNs, whch form the multcast tree, would be determned accordng to the followng approxmaton algorthm: parents can be chosen as RN f one of ther chldren has a fewer number of parents. When presented wth a number of RNs, the node wth the largest number of chldren s selected. Then, the elected RN together wth ts chldren s removed from the tree and the prevous two steps would be repeated untl all nodes at level + 1 are removed. Channel assgnment for MCM, on the other hand, can be performed by ether of the followng two algorthms reported n Zeng et al. (2010: the top-down Ascendng Channel Assgnment (ACA assgns channels to levels startng from zero. When t runs out of channels, the algorthm re-assgns channel zero to the nodes of the next level and ths process repeats. The shortcomngs assocated wth ACA are as follows; under-utlsaton may occur n case the number of sblng nodes s not equal n the whole tree. Also, ACA s prone to the hdden channel problem (Lm et al., Furthermore, for almost dagonal trees, ACA s subject to the same shortcomngs as those of LCAs (Zeng et al., The second channel assgnment scheme, referred to as Heurstc Channel Assgnment (HCA draws on the channel separatng concepton, whch ndcates the dsparty between two channel numbers. For example, the separaton between channels 2 and 5 s 3. In ths method when a channel s assgned to node u, t should mnmse the sum of squares of nterference factor (Dng et al., 2008 between node u and all nodes v n ts neghbourhood. However, snce t has only account sngle-hop neghbours, the hdden channel problem may occur. The hdden channel problem of Zeng et al. (2010 has motvated the work n Nguyen and Nguyen (2008, extended later n Nguyen and Nguyen (2009a. To reduce nterference, a ftness functon has been proposed to evaluate assgnment of channel c to node v. The hdden channel problem s dealt wth by factorng the channel nformaton of nodes wthn two hops nto the objectve functon. However, t s assumed n Nguyen and Nguyen (2008, 2009a that the multcast tree s readly avalable. It also reles on heavy broadcast message exchanges. Another evaluaton functon s utlsed n Yn et al. (2007; n partcular, the assgnment of channel c to node v s evaluated based on the probablty of packet transmsson by neghbourng nodes of v on channel c. However, the specfcs of the computaton of ths probablty are not elaborated n Yn et al. (2007. In Cheng and Yang (2008a, 2008b, 2011 three methods based on the GA, SA and the Tabu search technque have been proposed. In these methods, each multcast tree s represented by a two-dmensonal array (chromosome, the rows of whch determne a path from multcast source S to the recever R. Bascally, a chromosome has K rows for K recevers. The number of columns n the chromosome s equal to the maxmum path length from S to the multcast group members. The representaton of the chromosomes s based on the IDs of the nodes lyng along the path from sender S to the multcast group members. In ths method, chromosomes are created as follows: the algorthm begns from the multcast source node S. A one-hop neghbour of S s randomly selected and ts ID s nserted nto the chromosome. Ths process contnues tll t

5 On channel assgnment and multcast routng n mult-channel mult-rado wreless mesh networks 229 reaches a recever R. Thus, a row of chromosomes s made. The same process s also run for the next recever (wthn the next row. After constructng the multcast tree for each row of the chromosome, whch s essentally a path from S to R, the channels are sequentally assgned to the edges. The followng dsadvantages can be ponted out for these methods; frst, assgnment of the channels to each row of the chromosome s subject to smlar shortcomngs as those of the LCA algorthm;.e., the hdden channel problem s not accounted for. Second, the nodes wthn the same level n the multcast tree are prone to nterference. Fnally, f the number of channels s more than the number of multcast tree levels, some channels wll not be used at all. The methods proposed n Zeng et al. (2007, 2010, Cheng and Yang (2008a, 2008b, 2011, Nguyen and Nguyen (2008, 2009a and Yn et al. (2007 ether construct a multcast tree or have assumed the tree s already constructed and then channels are assgned. In Nguyen and Nguyen (2009, however, the channel assgnment s supposed to be done a pror and nstead, the multcast tree should be constructed. More specfcally, a centralsed multcast tree algorthm, namely, Mnmum Number of Transmsson (MCMNT, has been proposed, whch ams to mnmse the number of packets coped on to dfferent channels n each node. The objectve s to construct a tree wth mnmum cost usng ether the Djkstra or the Bellman_Ford algorthm. In Lm et al. (2009, a dstrbutng bottom-up approach has been proposed to construct a multcast tree and to establsh channel-rado assocatons. Intally, an approxmaton algorthm s used to generate the mnmum RN set and to construct the multcast tree; each node dentfes ts two-hop neghbours wth the least number of parents. Then, some of the parents of the two-hop neghbours wth fewer chldren are canddate. Canddates wth the best lnk qualty wll be selected. Afterwards, the selected nodes and all of ther chldren are removed from the sngle-hop and two-hop nodes lsts. Ths algorthm contnues untl both sngle-hop and two-hop neghbours lst are empty. However, t s subject to both the hdden channel problem and heavy broadcast message exchanges for placng nodes across levels. Table 1 recaptulates our revew of the exstng methods wth respect to ther underlyng deas for both multcast tree constructon and channel assgnment. In sum, the major drawback wth the exstng methods s that they have consdered channel assgnment and multcast tree constructon n the form of two ndependent ssues and thus, have taken on an essentally sequental approach to a jont problem. In partcular, the cross-nteracton between multcastng at the network and channel assgnment at the MAC layer s not accounted for n Zeng et al. (2007, Cheng and Yang (2008a, 2008b, 2011 and Lm et al. (2009 and n some methods, t s even assumed that the soluton for these two sub-problems s pre-calculated (e.g., Nguyen and Nguyen (2008, 2009a, 2009b and Yn et al. (2007. Ths s whle the cross-layer desgn forms an ntegral part of a successful WMN-based mplementaton, as has been extensvely and methodcally argued n Peng et al. (2007. Moreover, all the revewed schemes, except of course for Lm et al. (2009, are bult around a centralsed perspectve and ther heurstc mentalty s essentally ncapable of provdng the optmal soluton. Table 1 Exstng multcast desgns for Mult-Channel Mult-Rado Wreless Mesh Networks Ref. Zeng et al. (2007, 2010 Nguyen and Nguyen (2008, 2009a Yn et al. (2007 Cheng and Yang (2008a, 2011 Nguyen and Nguyen (2009b Lm et al. (2009 Multcast tree constructon A centralsed heurstc bottom-up algorthm whch utlses BFS MCM: A centralsed approxmate top-down algorthm Multcast tree s assumed to be constructed a pror Multcast tree s assumed to be constructed a pror A centralsed scheme based on GA, SA and TS technques MCMNT: A heurstc method for computng edge cost, coupled wth mnmum cost tree constructon usng ether the Djkstra or the Bellman-Ford algorthms An mproved verson of MCM whch also factors n the lnk qualty Channel assgnment LCA: An ascendng method An ascendng method A heurstc method based on channel separaton whch only consders sngle-hop nterference M4: An mproved verson of Zeng et al. (2010, whch also consders two-hop nterference A greedy channel assgnment that utlses BFS An ascendng method Channel assgnment s assumed to be performed a pror A heurstc algorthm Consderng the above weaknesses, n what follows, we manly focus on achevng the optmal soluton for jont channel assgnment and the multcast tree constructon problem n MCMR WMNs. For ths, we present two mathematcal BIP-based formulatons whch are nherently centralsed. The former solves the problem va a cross-layer desgn and the latter s a layered protocol whch provdes the optmal soluton for each sub-problem. 3 Mathematcal framework The mathematcal framework presented n ths paper s bascally a BIP formulaton, whch, as opposed to the exstng heurstc- or meta-heurstc-based schemes, guarantees the optmalty of the soluton. Gven a problem defnton n terms of a set of lnear equalty/nequalty constrants, a BIP model, as a varant of lnear programmng, determnes a way to obtan the best feasble soluton. From the geometrc vewpont, a feasble regon, n the form of a convex polyhedron, s

6 230 M. Jahanshah et al. defned wth lnear constrants. Wthn ths regon, provded that a feasble soluton exsts and also under the condton that the lnear objectve functon s bounded, the optmum result s always achevable on the boundary of optmal levelset by the maxmum/mnmum prncple for convex/concave functons (Vandenberghe, Gven the wholeness and properness of the set of constrants formulatng the problem of nterest, BIP guarantees global optmalty of the resultng soluton. 3.1 System model and assumptons In ths secton, we present the assumptons of our mathematcal formulatons for the problem of multcast tree constructon wth mnmal nterference. We have formulated our BIP models to fnd the MCT-based multcast tree. As for nterference computaton, we have adopted the protocol model descrbed n Hossan and Leung (2007 and Crchgno et al. (2008. In ths model, a gven transmsson from a node Src to a node Des s sad to be successful only f: the nodes dstance s less than the transmsson range no thrd node, located wthn the nterference range of the recevng node Des, s transmttng. The adopted model can further be refned to comply wth IEEE style MAC protocols;.e., the sendng node Src s also requred to be free of nterference, as t needs to receve the lnk layer acknowledgement from the recevng node Des. Specfcally, any node Temp, whch s wthn the nterference range of Src or Des, should not be transmttng. Other assumptons nclude: all mesh routers are dstrbuted randomly on a plane. Each router s equpped wth multple rado nterfaces and the number of rados s not more than that of the avalable non-overlappng channels. All rado nterfaces on wreless routers make use of omn-drectonal antennas and have dentcal transmsson/nterference ranges. 3.2 Proposed methods As prevously mentoned, n the exstng methods the crossnteracton between multcast at the network and channel assgnment at the MAC layer has not been consdered. Therefore, n ths secton, we present a BIP-based formulaton n whch both multcast tree constructon and channel assgnment problems are solved conjontly. We called ths method the cross-layer desgn. Afterwards, we wll propose another formulaton, namely the Sequental desgn, n whch the optmal soluton for each sub problem s provded. Before we present the formulatons, common defntons for both methods should be descrbed. Tables 2 and 3 show the common defned sets and parameters for both proposed sequental and cross-layer methods. Necessary descrptons are ncluded n the tables too Proposed cross-layered method In ths secton, the requred varables used wthn the BIP cross-layer model followed by the defned constrants are descrbed. Table 2 Common defned sets for both sequental and cross-layer methods Nodes = { N1, N2, N3,, N n } mesh routers ChannelLst = { C1, C2, C3,, C c } channels MultcastSource = { N}, N Nodes multcast sources MultcastTarget = { T1, T2, T3,.., Tt}, T Nodes multcast targets Rado = { R1, R2, R3,.., R R } mesh routers, rados R( N, N Nodes node N s rados Table 3 Common predefned parameters for both Sequental and cross-layer methods G = ( V, E /* A drected graph representng a mult-channel mult-rado wreless mesh network*/ R T /* Transmsson range */ RI = q RT, q 1 /* Interference range */ dsrc, Des, Src, Des Nodes /* Eucldean dstance between mesh node Src and Des */ UDG ( Src, Des, /* A bnary neghborhood Src, Des Nodes matrx named unt dsk graph whch s computed usng R T and dsrc, Des and specfes whether there s a lnk between Src and Des or not */ R( Src, Src Nodes /* Number of rados for node IsNegh( Src, Des, Temp1, Temp2, Src, Des, Temp1, Temp2 Nodes Defned varables Before presentng the BIP cross-layer model for the jont multcast tree constructon and channel assgnment problem, necessary varables should be ntroduced. In what follows, the varables are dscussed. A bnary varable F C, SR, Src where Src Nodes, SR Rados and C ChannelLst s defned to determne whether or not the channel C s assgned to rado SR of node Src. A bnary varable Lnk( Src, SR, Des, DR, C such that Src, Des Nodes, SR, DR Rados and C ChannelLst s defned to determne whether or not a lnk between rado SR of node Src and rado DR of node Des s establshed on channel C. Two non-negatve varables InputLnk(Src and OutputLnk(Src wth the followng defntons are requred to determne the number of ncomng/outgong lnks to/from mesh router Src, respectvely. InputLnk( Src Lnk( Temp, TR, Src, SR, C = C SR TR Temp OutputLnk( Src Lnk( Src, SR, Temp, TR, C = C SR TR Temp Src */ /* A bnary parameter whch determnes whether Temp1 or Temp2 s n nterference range of Src or Des or not */ (1 (2

7 On channel assgnment and multcast routng n mult-channel mult-rado wreless mesh networks 231 where Src, Temp Nodes, TR, SR Rados, C ChannelLst. Also, a non-negatve varable Interference(Src, SR, Des, DR, C such that Src, Des Nodes, SR, DR Rados, C ChannelLst s requred to determne the number of nterferng lnks wth lnk(src, SR, Des, DR, C. SIL s another non-negatve varable whch stands for Sum of Interferng Lnks and computes the number of lnks wthn the nterference range of Src or Des on channel C. Ths varable s manly ntroduced to account for the hdden channel problem: SIL( Src, SR, Des, DR, C = [ IsNegh( Src, Des, Temp1, Temp2 Temp2TR2Temp1 TR1 (3 Lnk( Temp1, TR1, Temp2, TR2, C Lnk( Src, SR, Des, DR, C ] where Src, Des Temp1, Temp2, Nodes, SR, DR, TR1, TR2 Rados, C ChannelLst. TotalInterference s a non-negatve varable whch determnes total nterference across the multcast tree: TotalInterference = Interference( Src, SR, Des, DR, C (4 Des DR Src SR C where Src, Des Nodes, SR, DR Rados, C ChannelLst. Fnally we defne a non-negatve varable whch determnes the number of lnks formng the multcast tree: TotalLnks = (5 Des DR Src SR C Lnk( Src, SR, Des, DR, C where Src, Des Nodes, SR, DR Rados, C ChannelLst Defned constrants In what follows the requred constrants n cross-layer desgn are ntroduced. From the multcast tree defnton, the number of all ncomng lnks to a node except for the multcast source and multcast target should be at most 1. Constrant (6 satsfes ths property. InputLnk( Src 1, Src Nodes\( MultcastSource, MultcastTarget. As s shown n constrant (7, n a multcast tree wth mult-rado nodes, the number of all ncomng lnks to and outgong lnks from a node except for the multcast source and multcast target should be at most equal to the number of rados per each node. InputLnk( Src + OutputLnk( Src R( Src, Src Nodes\( MultcastSource, MultcastTarget. It s obvous that there are no ncomng lnks to a multcast source. Constrant (8 ensures ths feature. InputLnk( Src = 0, Src MultcastSource. (6 (7 (8 In a multcast tree, as shown n constrant (9, the number of outgong lnks from the multcast source should be at most equal to the number of multcast targets. OutputLnk( Src MultcastTarget, Src MultcastSource. The multcast source must have at least one outgong lnk. Constrant (10 guarantees ths property. OutputLnk( Src 1, Src MultcastSource. (9 (10 Also, all multcast targets should have just one ncomng lnk. Equalty constrant (11 ensures ths characterstc. InputLnk( Src = 1, Src MultcastTarget. (11 Smlarly, the number of outgong lnks from a multcast target should be exactly zero (constrant 12. OutputLnk( Src = 0, Src MultcastTarget. (12 In MCMR WMNs, every rado of a node should be used at most once for ether nput or output purposes. Constrant (13 mplements ths feature. Des DR C Lnk( Src, SR, Des, DR, C + Lnk( Des, DR, Src, SR, C 1 (13 Des DR C Src, Des Nodes, SR, DR Rados, C ChannelLst. Inequaltes (14 and (15, together, defne an f and only f constrant;.e., f a node has an ncomng edge, then t defntely has an outgong edge as well and vce versa. OutputLnk( Src InputLnk( Src, Src Nodes (14 OutputLnk( Src InputLnk( Src, Src Nodes. (15 R( Src In MCMR WMNs, every channel C should be assgned to rados of a node at most once. Inequalty constrant (16 guarantees ths property. Des DR SR Lnk( Src, SR, Des, DR, C + Lnk( Des, DR, Src, SR, C 1 (16 Des DR SR Src, Des Nodes, SR, DR Rados, C ChannelLst. The aforementoned varable Interference(Src, SR, Des, DR, C s computed as equaton (17. Interference( Src, SR, Des, DR, C = Lnk( Src, SR, Des, DR, C SIL( Src, SR, Des, DR, C. (17

8 232 M. Jahanshah et al. Thus, f Lnk( Src, SR, Des, DR, C s establshed, then ts nterference s equal to the varable SIL; otherwse, t s zero. Constrants (18 (20 correspond to the multplcaton of the two varables Lnk and SIL n LP terms. Interference( Src, SR, Des, DR, C ( Nodes ( Nodes 1 /2 Lnk( Src, SR, Des, DR, C Interference( Src, SR, Des, DR, C SIL( Src, SR, Des, DR, C Interference( Src, SR, Des, DR, C SIL( Src, SR, Des, DR, C Nodes 1 Nodes 2 (1 Lnk( Src, SR, Des, DR, C Src, Des Nodes, SR, DR Rados, C ChannelLst. (18 (19 (20 Constrants (21 and (22 stpulate the lnk establshment condtons between two nodes Src and Des: they are located wthn the transmsson range of each other a common channel s assgned to both rado SR of Src and rado DR of Des. Lnk( Src, SR, Des, DR, C d Lnk( Src, SR, Des, DR, C R Lnk( Src, SR, Des, DR, C F Src, Des C, SR, Src = Lnk( Src, SR, Des, DR, C F T C, DR, Des (21 (22 where Scr, Des Nodes, SR, DR Rados, C ChannelLst. Fnally, snce the nterference s computed from the establshed lnks, the objectve of ths model can be defned: Mnmse( TotalLnks + TotalInterference. Altogether, the BIP formulaton for cross-layer desgn s modelled as follows: Mnmse( TotalLnks + TotalInterference Subject to: InputLnk( Src 1, Src Nodes\( MultcastSource, MultcastTarget InputLnk( Src + OutputLnk( Src R( Src, Src Nodes \( MultcastSource, MultcastTarget InputLnk( Src = 1, Src MultcastTarget OutputLnk( Src = 0, Src MultcastTarget Des DR C Lnk( Src, SR, Des, DR, C + Lnk( Des, DR, Src, SR, C 1, Des DR C Src, Des Nodes, SR, DR Rados, C ChannelLst OutputLnk( Src InputLnk( Src OutputLnk( Src InputLnk( Src, Src Nodes R( Src Des DR SR Lnk( Src, SR, Des, DR, C + Lnk( Des, DR, Src, SR, C 1, Des DR SR Src, Des Nodes, SR, DR Rados, C ChannelLst ( ( Interference( Src, SR, Des, DR, C Nodes Nodes 1 /2 Lnk( Src, SR, Des, DR, C Interference( Src, SR, Des, DR, C SIL( Src, SR, Des, DR, C Interference( Src, SR, Des, DR, C SIL( Src, SR, Des, DR, C ( ( Nodes Nodes 1 / 2 (1 Lnk( Src, SR, Des, DR, C Src, Des Nodes, SR, DR Rados, C ChannelLst Lnk( Src, SR, Des, DR, C d Lnk( Src, SR, Des, DR, C R Lnk( Src, SR, Des, DR, C F Src, Des T C, SR, Src = Lnk( Src, SR, Des, DR, C F C, DR, Des Src, Des Nodes, SR, DR Rados, C ChannelLst Proposed Layered method In ths desgn, say Sequental method, we model the overall problem as two separated sub-problems whch are solved n sequence. Clearly, n the frst phase of the Sequental method, a MCT-based optmal multcast tree s constructed. Afterwards, the resultant tree constructed from the frst phase s fed to the channel assgnment phase as nput. Hereafter, we refer to the multcast tree constructon model as the MT_Model. Smlarly, the channel assgnment model s referred to as the CA_Model. The algorthm of the layered desgn s shown n Fgure 1. In two sub-sectons and , two sub-problems of ths approach are dscussed n detal. Fgure 1 Algorthm of the proposed Sequental method InputLnk( Src = 0, Src MultcastSource OutputLnk( Src MultcastSource, Src MultcastSource OutputLnk( Src 1, Src MultcastSource

9 On channel assgnment and multcast routng n mult-channel mult-rado wreless mesh networks Multcast tree constructon sub-problem Defned varables In ths secton, the requred varables wthn the MT_Model of the proposed Sequental method are ntroduced as follows: A bnary varable s needed to ndcate whether there s a lnk between rado SR of node Src and rado DR of node Des or not. We defne ths varable as: Lnk( Src, SR, Des, DR, Src, Des Nodes, DR, SR Rados. Also, a non-negatve varable whch determnes the numbers of ncomng lnks to every mesh router s requred. For ths, we defne the varable InputLnk(Src wth the equaton (23. InputLnk( Src = Lnk( Des, DR, Src, SR SR Des DR Src, Des Nodes, DR, SR Rados (23 Smlarly, we need another non-negatve varable whch determnes the number of outgong lnks from every mesh router. Thus, we defne the varable OutputLnk(Src wth defnton (24. OutputLnk( Src = Lnk( Src, SR, Des, DR, SR Des DR Src, Des Nodes, DR, SR Rados (24 Fnally, the non-negatve varable Total_Lnk s requred to determne the total number of lnks formng the multcast tree: Total_ Lnk = Lnk( Src, SR, Des, DR, Src SR Des DR Src, Des Nodes, DR, SR Rados Defned constrants (25 In ths sub-secton, the requred constrants for the multcast tree constructon sub-problem of the proposed Sequental method are dscussed; In case of the multcast tree, the number of all the nput lnks to a node except multcast source and multcast target should be at most 1. Constrant (26 ensures ths property. InputLnk( Src <= 1 Src Nodes\( MultcastSource,MultcastTarget. (26 In MCMR WMNs, the sum of all ncomng lnks to and outgong lnks from a node except multcast source and multcast target should be at most equal the number of rados. Constrant (27 guarantees ths feature. InputLnk( Src + OutPutLnk( Src <= R( Src, Src Nodes\( MultcastSource,MultcastTarget. (27 In MCMR WMNs, f two nodes are located n the transmsson range of each other, then one lnk can be establshed between them (constrant (28. Lnk( Src, SR, Des, DR <= UDG( Src, Des, Src, Des Nodes, DR, SR Rados. (28 From the multcast tree defnton, the multcast source node has no ncomng lnks. The equalty constrant (29 satsfes ths property. InputLnk( Src = 0, Src MultcastSource (29 Smlarly, the number of outgong lnks from the multcast source node should be at most equal to the number of multcast targets. Ths feature s ensured by constrant (30. OutputLnk( Src <= MultcastTarget, Src MultcastSource. (30 Furthermore, the number of outgong lnks from multcast source node should be at least one(constrant 31. OutputLnk( Src >= 1, Src MultcastSource (31 The number of ncomng lnks to every multcast target should be exactly one: InputLnk( Src = 1, Src MultcastTarget. (32 Equalty constrant (33 ensures the multcast targets have no outgong lnks. OutputLnk( Src = 0, Src MultcastTarget. (33 The proposed formulaton should occur wth guarantees of one-hop loops n the multcast tree. Constrant (34 ensures ths nherent feature of the constructed tree. UDG( Src, Temp Lnk( Src, SR, Temp, TR SR TR + Lnk( Temp, TR, Src, SR 1 (34 SR TR UDG( Src, Temp Src, Temp Nodes, TR, SR Rados. In MCMR WMNs, every rado of a node has to form at most one lnk: UDG( Src, Des Lnk( Src, SR, Des, DR Des DR + Lnk( Des, DR, Src, SR 1 (35 Des DR UDG( Src, Des. Constrants (36 and (37 explan an f and only f condton. That s, f a node has one or more outgong lnks, then t should have an ncomng lnk and also f t has an ncomng lnk, then t wll have at least one outgong lnk: OutputLnk( Src >= InputLnk( Src, Src Nodes \( MultcastSource, MultcastTarget OutputLnk( Src <= InputLnk( Src, rados Src Nodes\( MultcastSource, MultcastTarget. (36 (37

10 234 M. Jahanshah et al. And fnally, the objectve of the multcast tree constructon sub-problem s: MnmseTotal _ Lnk. The BIP model for ths sub-problem of the proposed layered desgn s summarsed as follows: MnmseTotal _ Lnk Subject to: InputLnk( Src <= 1, Src Nodes\( MultcastSource, MultcastTarget ( InputLnk( Src + OutPutLnk( Src <= R( Src, Src Nodes\( MultcastSource, MultcastTarget Lnk( Src, SR, Des, DR <= UDG( Src, Des, Src, Des Nodes, DR, SR Rados InputLnk( Src = 0, Src MultcastSource OutputLnk( Src <= MultcastTarget, Src MultcastSource OutputLnk( Src >= 1, Src MultcastSource InputLnk( Src = 1, Src MultcastTarget OutputLnk( Src = 0, Src MultcastTarget UDG( Src, Temp Lnk( Src, SR, Temp, TR SR TR + Lnk( Temp, TR, Src, SR 1 SR TR UDG( Src, Temp, Src, Temp Nodes, TR, SR Rados UDG( Src, Des Lnk( Src, SR, Des, DR Des DR + Lnk( Des, DR, Src, SR 1 Des DR UDG( Src, Des OutPutLnk( Src >= InputLnk( Src, Src Nodes\( MultcastSource, MultcastTarget OutPutLnk( Src <= InputLnk( Src, rados Src Nodes\( MultcastSource, MultcastTarget Channel assgnment sub-problem After solvng the MT_Model and the CA_Model, here t s necessary to defne a bnary parameter Lnk_ whch s computed usng the bnary varable Lnk( Src, Des as output of the frst phase (MT_Model. The parameter Lnk_ s set to one f Lnk ( Src, Des has been establshed and zero otherwse Defned varables We defne four varables wthn the CA_Model of the proposed Sequental desgn as follows: A non-negatve varable Sum of Interferng Lnks (SIL s needed to compute the number of lnks wthn the nterference range of the source or destnaton of a lnk: SIL( Src, Des, C = Lnk_( Src, Des ( IsNegh( Src, Des, Temp1, Temp2 Temp1Temp 2 Lnk _( Temp1, Temp2 Assgn( Temp1, Temp2, C Assgn( Src, Des, C (38 where Src, Des, Temp1, Temp2 Nodes, C ChannelLst. Ths varable consders the hdden channel problem as well. A bnary varable Assgn( Src, Des, C such as Src, Des Nodes and C ChannelLst, s needed to determne whether C s assgned to the establshed lnk between two nodes Src, Des or not. The non-negatve varable Interference( Src, Des, C, denotes the number of nterferng lnks wth lnk (Src, Des, C and s computed as equaton (39. Interference ( Src, Des, C = Assgn( Src, Des, C SIL ( Src, Des, C. (39 Fnally, the non-negatve varable Total_Interference s defned to compute the overall nterference n the multcast tree: Total _ Interference = Interference( Src, Des, C Src Des C (40 Src, Des Nodes, C ChannelLst Defned constrants The requred constrants for the CA_Model of the proposed Sequental method are dscussed as follows: Just one channel should be assgned to every lnk n MCMR WMNs (constrant (41. C Assgn( Src, Des, C = Lnk _( Src, Des, Src, Des Nodes, C ChannelLst (41 If a lnk s establshed and channel C s assgned to t, then ts nterference s equal to varable SIL; otherwse, t s zero. Constrants (42 (44, correspond to the multplcaton of the two varables Lnk and SIL n LP terms. UDG( Src, Des Lnk _( Src, Des Interference ( Src, Des, C UDG( Src, temp1 UDG ( Src, temp1 1 temp 1 temp1 + UDG( Des, temp2 UDG( Des, temp2 1 temp 2 temp 2 Assgn ( Src, Des, C UDG ( Src, Des Lnk _( Src, Des (42

11 On channel assgnment and multcast routng n mult-channel mult-rado wreless mesh networks 235 UDG ( Src, Des Lnk_( Src, Des Interference ( Src, Des, C SIL ( Src, Des, C UDG ( Src, Des Lnk _( Src, Des, Src, Des Nodes, C ChannelLst temp1 (43 UDG ( Src, Des Lnk _( Src, Des Interference( Src, Des, C ( SIL( Src, Des, C UDG ( Src, temp1 UDG( Src, temp1 1 + UDG( Des, temp2 temp1 temp 2 UDG ( Des, temp2 1 (1 Assgn( Src, Des, C temp 2 UDG( Src, Des Lnk _( Src, Des Src, Des, temp1, temp2 Nodes, C ChannelLst. And fnally, the objectve of the CA_Model s: MnmseTotal _ Interference (44 Therefore, the BIP model for channel assgnment subproblem of the layered scheme s represented as follows: Mnmse Total_Interference Subject to: Assgn( Src, Des, C = Lnk_( Src, Des, C Src, Des Nodes, C ChannelLst UDG ( Src, Des Lnk _( Src, Des Interference( Src, Des, C UDG( Src, temp1 UDG ( Src, temp1 1 temp 1 temp1 + UDG( Des, temp2 UDG ( Des, temp2 1 temp2 temp2 Assgn( Src, Des, C UDG( Src, Des Lnk_( Src, Des 4 Connectvty and loop consderatons We tend to demonstrate the mmunty of our formulaton to loop formatons as well as to ensure connectvty that s preserved wth the resultng channel-rado assgnments. Our essental reason for the worst case s also evdently applcable to the average scenaro. 4.1 Connectvty We examne the multcast tree connectvty n the proposed cross-layer method. A smlar lne of reasonng s also applcable to the case of our Sequental desgn. Defnton 1. Connectvty s satsfed only f there exsts a path between the multcast source and all multcast group members. Property 1. The BIP formulaton gven n Secton 3 guarantees connectvty across the multcast tree. Proof: We demonstrate the noton of connectvty usng the Unt Dsk Graph (UDG depcted n Fgure 2(a wthout loss of generalty. In ths graph, each lnk between two nodes ndcates that they are located wthn the transmsson range of each other. Two nter-lnked nodes would be able to communcate f only dentcal channel numbers get assgned to one of ther rados. MS denotes the multcast source and MT stands for the multcast target. Constrant (11 ensures that MT has exactly one ncomng lnk, as llustrated n Fgure 2(b. Constrants (14 and (15 stpulate that n case a mesh node has an outgong lnk, t ought to have an ncomng lnk as well. Inequalty (6 requres that only one of the lnks (1, 2 or (3, 2 be establshed as demonstrated n Fgure 2(c. On the other hand, constrant (10 warrants that MS has at least one outgong lnk. Therefore, mesh node 1 has to be necessarly assocated wth an output and the outgong lnk from node 3 should be removed (Fg. 2(d. Hence, there exsts a path from MS to MT. Fgure 2 Multcast tree connectvty UDG ( Src, Des Lnk_( Src, Des Interference( Src, Des, C SIL( Src, Des, C UDG( Src, Des Lnk_( Src, Des Src, Des Nodes, C ChannelLst UDG ( Src, Des Lnk_( Src, Des Interference( Src, Des, C ( SIL( Src, Des, C UDG( Src, temp1 UDG( Src, temp1 1 temp1 temp1 + UDG( Des, temp2 UDG( Des, temp2 1 temp2 temp2 (1 Assgn( Src, Des, C UDG( Src, Des Lnk _( Src, Des, Src, Des, temp1, temp2 Nodes, C ChannelLst 4.2 Loop occurrence We nvestgate the correctness of our proposed formulaton for the proposed cross-layer method n terms of the loop formaton ssue through the followng property. Here agan we have consdered the worst case scenaro. A smlar analyss apples to the Sequental scheme. Property 2. The multcast tree assocated wth the BIP formulaton gven n Secton 3 s guaranteed to be loopfree.

12 236 M. Jahanshah et al. Proof: Gven the UDG depcted n Fgure 2(a, we assume that there exsts a loop, say (1, 2, 3, 1 wthn the multcast tree, as shown n Fgure 3(a. Constrants (9 and (10 ensure that MS has at least one outgong lnk, as shown n Fgure 3(b. Inequalty (6 requres that lnk (3, 1 be removed from the confguraton (Fg. 3(c. Also, constrants (14 and (15 warrant that node 3 has no ncomng lnk (Fg. 3(d. Fnally, Constrant (11 stpulates that the MT has exactly one ncomng lnk (Fg. 3(e. 4-rado networks leads to dsconnecton of the wreless mesh network, ther correspondng results have not been shown n Fgures 4(a and (b. Fgure 4 Impact of channel number varatons on the overall nterference (see onlne verson for colours Fgure 3 Multcast tree loop preventon 5 Smulaton results We mplemented our BIP framework usng CPLEX 11 (for more nformaton about CPLEX, see Ilog Inc. (2011. Further, to evaluate the performance of the proposed methods, several smulatons have been conducted. Smulaton 1 was conducted to observe the mpact of channel number varatons on the overall nterference n our cross-layer desgn, as compared wth LCA, MCM (Zeng et al., 2007, 2010 and methods based on the GA, SA and the Tabu search (Cheng and Yang, 2008a, 2008b, In smulaton 2, the proposed cross-layer and Sequental methods are evaluated n terms of optmalty, say overall nterference, on the one hand and complexty, say requred tme to solve the multcast problem and memory consumpton, on the other hand. Our Sequental method s compared wth methods based on LCA, MCM, GA, SA and TS n smulaton 3. Smulaton 4 s dedcated to compare the resultant nterference for dfferent methods consderng rado number varatons. All smulatons were conducted on a dense random generated UDG n whch the number of nodes and the sze of multcast recevers set are 30 and 13 respectvely. Smulatons are descrbed as follows. 5.1 Smulaton 1 In ths smulaton, we study the resultant nterference n the proposed cross-layer desgn as compared wth LCA and MCM, as well as wth methods based on GA, SA and TS. Recall from secton 1 that the number of channels n MCMR WMNs should be more than or equal to the number of rados. Therefore, for a network wth n-rado nodes, a set of smulatons for n through m channels should be conducted (m n, where m s the pont at whch nterference reaches zero. Fgures 4(a through 4(f show the resultant nterference for dfferent methods, when the mesh nodes are equpped wth dfferent numbers of rados. In Fgures 4(a (e, every curve s the result of a smulaton set, not just one smulaton. In other words, every data pont shows the result of a smulaton n a smulaton set. Each data pont assocated wth methods based on LCA, MCM, GA, SA and TS n ths smulaton s the average of 20 tmes smulatons. Snce utlsng the methods based on GA, SA and TS n 3- and

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