Optimal Multicast in Multi-Channel Multi-Radio Wireless Networks

Transcription

1 Optmal Multcast n Mult-Channel Mult-Rado Wreless Networks Aay Gopnathan, Zongpeng L, Carey Wllamson Department of Computer Scence, Unversty of Calgary {agopnat,zongpeng,carey}@cpsc.ucalgary.ca Abstract Recent advances n wreless technology have made t ncreasngly feasble to equp wreless nodes wth multple rados, thereby allowng each rado to explot channel dversty n the form of orthogonal, non-overlappng transmsson spectrums. Mult-channel operaton mtgates nterference, but at the same tme rases new challenges for network optmzaton, n terms of udcous channel assgnment for effcent bandwdth utlzaton. Whle prevous research mostly studes optmzng channel assgnment for uncast, we focus nstead on multcast, whch s an effcent mechansm for one-to-many data dssemnaton. We derve a model for optmal multcast n mult-channel mult-rado wreless networks under the assumpton that channel assgnment s statc. Our model employs network codng as the multcast mechansm of choce, and explots the broadcast nature of omndrectonal antennas for effcent bandwdth utlzaton. Based on the model derved, we formulate optmal multcast as a lnear nteger program. Two accompanyng solutons are proposed: a greedy channel assgnment scheme and an mproved teratve scheme nspred by prmal-dual algorthm desgn. The effectveness of the two schemes are emprcally examned through smulaton studes, and are compared to results obtaned from solvng the nteger program as well as ts lnear programmng relaxaton. Fnally, we present an alternate model for optmal multcast under the assumpton that transmsson frequences are not fxed dvsons of the usable spectrum. I. INTRODUCTION Wreless networks have become ncreasngly prevalent as the preferred choce for network connectvty n many mltary and cvl applcatons. For nstance, wreless networks provde moblty-frendly network access, and wreless LANs and wreless mesh networks often consttute a cost-effectve last mle connecton to the Internet. A fundamental problem that lmts the performance of wreless networks s nterference. Despte the advertsed transmsson speeds for wreless networks, the actual goodput achevable s approxmately halved when nterference effects and medum access contenton are taken nto account []. Interference can be caused by external sources, or by transmsson from nearby wreless nodes. As an example of the former, the IEEE 802.b and 802.g standards, whch operate n the 2.4 GHz spectrum, are susceptble to nterference caused by mcrowave ovens, bluetooth devces and even older cordless telephones. To counter the effects of nterference, channel dversty can be exploted. The IEEE 802. standard dvdes the usable spectrum nto a number of channels; for example, the GHz to GHz band s dvded nto 3 channels. Three of them (channels, 6 and ) occupy nonoverlappng frequency spectrums, and transmssons on these orthogonal channels do not nterfere. To take advantage of channel dversty, wreless nodes can be equpped wth multple wreless network cards or rados. Advances n wreless technology have demonstrated that multple rados per wreless node are a feasble and cost-effectve soluton [2]. Furthermore, recent research has shown that consderable performance gan can be acheved n mult-channel mult-rado wreless envronments [] [4]. Ths rases nterestng new challenges n the desgn of algorthms and mechansms that are capable of fully explotng channel dversty n mult-rado networks to optmze bandwdth usage, network throughput or routng cost. Prevous research n mult-channel mult-rado wreless networks has studed optmal uncast or mult-sesson uncast [3], [5], [6]. In contrast, we focus on the multcast problem nstead. Multcast s an effcent way of dssemnatng dentcal data to multple users and has numerous applcatons n both wrelne and wreless networks [7] [0]. Uncast and multcast dffer n terms of what consttutes a udcous channel assgnment scheme, whch assgns one of the avalable channels to each rado at each node. The set of avalable channels s a lmted resource, and an optmal channel-to-rado assgnment n the case of uncast may not necessarly be optmal for nterferencefree multcast routng. Consder the example n Fgure. Here, the source S needs to send nformaton to potental recevers T, T 2 and T 3. Each node s equpped wth 3 rados of unt capacty each, where separate rados are used to transmt and receve. The channels used for transmsson are ndcated next to each node. Channels are assgned so as to avod nterference, hence neghbourng nodes (ndcated by the dotted lnes) do not transmt on the same channel. Smlarly, nodes recevng transmssons from multple neghbours receve on dfferent channels as well. In Fgure (a), the channel assgnment s an optmal assgnment n the case of uncast from S to recever T, allowng recept of both bts a and b for a total throughput of 2 bts. Smlar channel assgnments can be made sequentally for uncast flows from S to T 2, and from S to T 3. However, these channel assgnments would not work for a network coded multcast sesson to all three recevers smultaneously. Explotng network codng for bandwdth effcency requres us to send the logcal XOR of bts a and b from node v to T 2, and from node T 2 to T and T 3. Usng the channel assgnment n Fgure (a), node v would not be able to lsten to nodes u and w concurrently snce both of these nodes transmt on channel 3, and thus ther transmssons would nterfere at v.

2 Secton IV. Secton V contans the alternate model for channel assgnment under the assumpton of flexble spectrum dvson. The paper s concluded n Secton VI. Fg.. An optmal channel assgnment scheme n the case of uncast s shown n (a). The flow of bts a and b are shown from S to T. Ths assgnment dffers from the optmal channel assgnment for multcast, shown n (b) The optmal channel assgnment for nterference-free multcast s nstead shown n Fgure (b). Node w transmts on channel 4 nstead of 3, allowng for nterference-free recepton at v. In ths paper, we seek to derve a complete mathematcal model of mult-channel mult-rado wreless networks, and formulate a mathematcal program for solvng the ont routng and channel assgnment problem for optmal multcast. We explot the prncples of network codng [], [2] for optmal multcastng, and develop routng constrants that explot the broadcast property of wreless networks for effcent bandwdth utlzaton. The result s a lnear nteger program that computes a ont channel assgnment and flow routng scheme, achevng maxmum end-to-end multcast throughput. Snce n practce nteger programs can be solved for small networks only, we desgn two algorthms for larger networks. In the frst algorthm, we relax the lmtatons mposed by channel and rado avalablty, and solve the resultng lnear program for an optmstc multcast flow. Channel assgnment then proceeds n a greedy fashon, n breadth-frst order along the computed flow. Then a new, feasble multcast flow s computed by pluggng the channel assgnment nformaton nto the optmal multcast nteger program and solvng the resultng lnear program. The second algorthm employs the frst one for generatng an ntal channel assgnment. Then t teratvely refnes the channel assgnment and the routng scheme based on the new soluton obtaned. Ths s nspred by prmal-dual algorthm desgn, where a par of prmal and dual solutons are teratvely refned untl convergence. In each teraton of the prmal-dual algorthm, the new solutons n the prmal (dual) provde drectons for updatng the dual (respectvely, prmal) n the next round [9]. We study these two algorthms through extensve smulatons, and show ther effcacy by comparng them wth results obtaned from solvng the nteger program and ts lnear programmng relaxaton. Fnally, we also present an alternate model for channel assgnment under the assumpton of flexble spectrum dvson. The rest of the paper s organzed as follows. Secton II dscusses related work. In Secton III, we derve a mathematcal framework to model multcast n mult-channel mult-rado wreless networks, and present an nteger program that maxmzes end-to-end throughput. We then proceed to outlne two channel assgnment algorthms. We show smulaton results n II. RELATED WORK Multcast s used for one-to-many dssemnaton of data, and s seen as an attractve data delvery mechansm, especally wth the ncreased prolferaton of applcatons such as multmeda streamng. Consequently, the multcast problem has receved a lot of attenton from the networkng research communty [7], [3] [5]. Computng optmal multcast routng (wthout network codng) s equvalent to the well-studed problem of fndng and packng Stener trees [7], [6]. The latter has however been shown to be NP-hard [7], though constant approxmaton algorthms exst usng LP roundng [8], prmal-dual schemes [9] and mnmum spannng tree heurstcs [20]. A more recent approach n the lterature s network codng, whch explots the encodable as well as replcable property unque to data flows. The fundamental result of network codng [], [2] states that a target multcast rate of d s achevable f and only f the rate d s a feasble uncast rate from the source to each and every recever. Usng ths result, multcast can be reduced to computng the optmal unon of uncast flows to each and every recever. The latter can be computed effcently usng sutably formulated lnear programs [9], [0], [2]. Mult-channel wreless networks have recently receved attenton from the research communty, mostly n the context of wreless mesh networks [22]. Ranwala et al. consdered the benefts of usng multple channels n wreless mesh networks [3]. Whle pror work consdered modfcatons to the IEEE 802. MAC protocol, Ranwala et al. focused on exstng IEEE 802. MAC protocols, and presented a load-aware channel assgnment algorthm. Ther algorthm s based on a greedy heurstc. Nevertheless, they show va smulatons that even 2 rados per node can lead to throughput mprovement by a factor of 8. Ranwala and Chueh [] extend ths work by proposng dstrbuted channel assgnment algorthms and demonstrate ts effectveness by performng experments on a 9-node network equpped wth commodty 802.a network cards. They fnd that n the context of long lved TCP flows, aggregate throughput can be 5 tmes hgher compared to sngle channel operaton. The semnal work of Gupta and Kumar [23] studed and characterzed the achevable capacty of wreless ad-hoc networks wth a sngle channel. Ther work s applcable to the mult-channel scenaro under the assumpton that rados are not able to swtch channels on a per packet bass. In subsequent work, Kyasanur and Vadya [24] extend the analyss of Gupta and Kumar and characterzed the capacty regon for mult-channel networks where rados are allowed to swtch channels. They fnd that the capacty of mult-channel multrado networks s hghly dependent on the rato between the number of rados and avalable channels. The channel assgnment problem s known to be NP-hard, and has receved much attenton n the lterature. Notably,

3 Brzeznsk et al. [25] study the problem from a graph theoretc pont of vew. They characterze nterference patterns usng lne graphs, and present a scheme that parttons the network usng local poolng, a greedy schedulng algorthm due to Dmaks and Walrand [26]. Brzeznsk et al. use local poolng to partton the network and show thereafter that smple dstrbuted channel assgnment and schedulng algorthms s suffcent to acheve maxmal throughput. However, the latter s true only for a restrcted class of network topologes. In contrast, Xng et al. [27] take an altogether dfferent approach and solve the channel assgnment problem usng codng theory. Specfcally, they use supermposed codes that are dstrbuted to nodes n the network. These nodes then attempt to choose a channel based on these codes. However, ths scheme suffers from the large dmenson of codes requred, and the lkelhood of falure of the code n hghly connected networks. Fnally, a more practcal approach s consdered by Ko et al. [4] n whch a dstrbuted greedy heurstc s used for channel assgnment. The heurstc entals that nodes contnuously try to choose channels that suffer the least nterference. They show that ths assgnment scheme s guaranteed to converge to an equlbrum whereby no node can mprove local nterference by swtchng channels. Experments performed usng IEEE 802.a/g network cards show a 50% mprovement n throughput. The use of mult-channels n mult-rado networks necesstates the desgn of new routng metrcs. Extendng the noton of the Expected Transmsson Count (ETX) metrc proposed by De Couto et al. [28] for mult-hop wreless routng, Draves et al. proposed the Weghted Cumulatve Expected Transmsson Tme (WCETT) [29] for mult-channel mult-rado wreless networks. The proposed routng metrc ncorporates the vagares of mult-channel routng n that t explctly takes nto account nterference effects, whch t tres to mnmze. The routng metrc was subsequently mplemented as a vrtual network drver, and experments show that n a prototype 2-rado network, WCETT acheves 89% hgher throughput than ETX, and n comparson wth shortest path routng, an mprovement of 254% n throughput s observed. The work of Kodalam and Nandagopal [6] as well as Alcherry et al. [5] s most relevant to the work presented n ths paper. Both form mathematcal models of mult-channel mult-rado networks. Interference due to transmssons on the same channel s consdered as a constrant, and lnear programmng s used to provde a frst approxmaton of the optmal flow route. Kodalam and Nandagopal study multcommodty routng, and provde heurstcs for both statc and dynamc channel assgnment. The channel assgnment algorthm scales the fractonal flow computed by the lnear program to ntegral values by multplcaton wth a large factor, whch s used to gude schedulng n whch a node s assgned a channel for a set number of tme slots dependent on flow values on that channel. A smlar flow scalng approach for schedulng and channel assgnment s presented by Alcherry et al., and n addton, the authors show that the throughput acheved usng ther method s wthn a constant factor of the optmal. In contrast to both papers, we study multcast routng n multchannel mult-rado wreless networks nstead. III. MULTICAST ROUTING AND CHANNEL ASSIGNMENT In ths secton, we defne our wreless network model and state our assumptons. We then proceed to model optmal multcast routng n mult-rado mult-channel wreless networks as an nteger program. Our nteger program s complete n that the constrants capture necessary and suffcent condtons to ensure nterference free routng assumng no schedulng s allowed,.e., we consder statc channel assgnment only n ths paper. Due to ntractablty of nteger programs n general, we further present two channel assgnment algorthms n ths secton. The frst s a smple greedy assgnment scheme, whle the second s a more sophstcated algorthm that teratvely makes mprovements on the resdual network of the flow graph to acheve hgher throughput. A. Prelmnares A wreless network s modeled as a graph G = (V, E), where V and E denote the set of nodes and edges respectvely. Nodes are equpped wth multple rados wth a unform coverage radus r. The set of rados n each node u s R(u), and each rado can be tuned to any channel n the set of channels K. The bandwdth on each rado s lmted, and we use c (u) to denote the capacty of the -th rado on node u. Each channel K s orthogonal to all other channels K, n the sense that transmssons on channels, K do not nterfere f. A node v s sad to be a neghbour of u f v s located wthn dstance r of u. The set of u s neghbours s denoted N(u), whle neghbours wthn two hops of u are n the set N 2 (u). An edge uv E f and only f v N(u). Edges are drected and symmetrc due to the unform radus of coverage property. Our am s to compute a multcast routng that s optmal n the sense that t maxmzes end-to-end throughput from some desgnated source node s, to the set of k multcast recevers T = {t,..., t k }. We abstract the dssemnaton of data from the source to the set of recevers as a network flow from the former to the latter. In realty, a unt of flow can be consdered to be an arbtrary unt of nformaton. For example, a unt of flow may correspond to a bt of nformaton, n whch case the capacty of a rado would ndcate the correspondng lmt of nformaton flow per tme unt, e.g. bts per second. For the purpose of computng the optmal flow, we model the flow rate on a lnk usng flow varables both wth and wthout an assocated channel usage, and we wll later defne how these varables are related. Recall that wth network codng, computng the optmal multcast flow s equvalent to computng the optmal unon of uncast flows from the source to each and every recever. We wll call these uncast flows conceptual flows [9], [2]. Conceptual flows coexst n the sense that they do not compete for bandwdth. We denote conceptual flow on a lnk as f t (.), whch represents the flow from the source to recever t T. A flow on a lnk uses one of many avalable channels, and hence f t (.) s the conceptual flow to recever t T usng channel K.

4 We wll use bnary varables to express how rados on a node are assgned to channels. Further, we wll dstngush rados that are transmttng data from those that are lstenng. The bnary varables x (u) and y (u) for the node u have the followng nterpretaton x (u) = y (u) = { rado on u transmts on channel, 0 otherwse { rado on u lstens on channel, 0 otherwse B. An Integer Programmng Formulaton Havng stated our model, we next need to clearly defne how the flow varables as well as bnary varables are related. The nter-dependence of these varous flow varables s crucal and together wth the constrants that capture the effect of nterference, completely characterzes the feasble flow regon. In ths secton, we wll state the requrements for a multcast flow that s nterference-free and optmal. We begn wth flow conservaton. To more succnctly express flow conservaton constrants, we ntroduce a vrtual, uncapactated drected lnk ts from every t T to s. For each conceptual flow, flow conservaton must therefore be observed at every node n the network, captured by the followng constrant: f t ( uv) () (2) f t ( vu) =0 t T, v V (3) The above states that the sum of all ncomng flow destned to each recever t on edges uv for all neghbours u of v must equal the sum of the conceptual flow on outgong lnks. Ths constrant must hold for all nodes n the network. Employng network codng, a multcast rate d can only be acheved f d s a feasble uncast rate to all recevers. Hence, n the optmal multcast flow, a rate of d must be acheved by each recever: f t ( ts) =d t T (4) The conceptual flow on a lnk uv can use one or more channels, and so: ft ( uv) =f t ( uv) uv E, t T (5) K Recall that conceptual flows do not compete for bandwdth [9], [2]. Hence, the actual outgong flow from any node on any channel s the maxmum of all conceptual flows usng that channel on the node s outgong lnks. Note that the max functon s non-lnear. However, the followng constrant captures our requrement n an equvalent fashon n the context of the optmzaton problem: c (u)x (u) t T, K, u V(6) f t ( uv) There are several thngs worth notng about constrant (6). Frst, (6) mplctly takes the wreless broadcast property nto consderaton. The broadcast property s unque to wreless networks, as transmsson to any node n the neghbourhood of u can be overheard by any node v N(u). Clearly, any multcast formulaton should seek to explot ths property. The constrant n (6) states that transmssons on channel can potentally flow to any neghbourng node, and further, the bandwdth utlzed by the broadcast s lmted to the capacty of the rado used for transmsson. Note also that (6) allows multple rados to be used to transmt on a sngle channel. Clearly, ths results n nterference and s undesrable. Nevertheless, we wll mpose further restrctons to ensure that at most one rado on a node can be assgned to a gven channel. The sum over all rados R(u) here then takes on a dfferent semantc, namely, regardless of the choce of rado used, the flow s upper bounded by the bandwdth provsoned on that rado. Fnally, the bnary varable x (u) s used to ensure that f a rado s used for any non-zero amount of outgong flow on channel, then that rado s assgned to that channel. Next, we relate ncomng flow to channel assgnment on the lstenng rado of a node. We wll use conceptual flow varables on ncomng lnks to state the followng requrement for channel assgnment on the recevng rado: c (u)y(u) t T, K, u V (7) f t ( vu) The above constrant states that to receve flow on a gven channel, there must be at least one rado that s tuned to lsten on that channel. In the case of heterogeneous capactes on rados, the flow across a lnk s bounded by the capactes of both the transmttng and recevng rado, and thus the true capacty s the mnmum of these two rados. Ths model works well n realty, snce the IEEE 802. standard specfes mechansms for dynamc rate shftng [30], thereby allowng the transmttng and recevng rados to agree on the acceptable data rate. Next we deal wth channel assgnment and node-rado constrants. Frst, a rado can ether be transmttng or lstenng, but never both: x (u)+y (u) K, R(u), u V (8) Second, the number of rados on any node s lmted, and so: x (u) R(u) u V (9) y(u) R(u) u V (0) Thrd, a node can only transmt (receve) on a gven channel usng a sngle rado to avod local nterference: x (u) K, u V () y(u) K, u V (2)

5 Fnally, we requre nterference-free transmsson. We wll mpose constrants based on the protocol nterference model [6]. The IEEE 802. RTS-CTS mechansm requres nodes to request clearance from the potental recever before transmttng. The protocol also acts as a vrtual carrer sensng mechansm, n that neghbourng nodes are capable of knowng when there s an mpendng transmsson by other nodes wthn nterference range. Ths substantally reduces the number of collsons. In ths model, neghbours of both the transmtter and ntended recpents should not transmt at the same tme on the same channel. The followng constrant captures ths behavour: (x (u)+ ) x (v) K, u V (3) v N 2(u) Ths fnal constrant requres that for a node u to transmt on a gven channel, no other node v wthn two hops of u s allowed to transmt on that channel. Observe that ths constrant can easly be modfed to work under the prmary nterference model [5], [25] as well by replacng the set N 2 (u) wth N(u). Our obectve s to maxmze the target multcast rate d. We now present our nteger program for optmzng end-to-end multcast throughput n ts entrety: f t( uv) ( x (u)+ Maxmze d (4) Subect To f t( ts) =d f t( vu) = 0 ft ( uv) =f t( uv) K ft ( uv) c (u)x (u) ft ( vu) c (u)y(u) x (u)+y(u) x (u) R(u) y(u) R(u) x (u) y(u) v N 2 (u) x (v) ) f t( uv),f t ( uv) 0 x (u) {0, },y (u) {0, } C. Lnear programmng relaxaton t (4a) t, v (4b) uv, t (4c) t,, u (4d) t,, u (4e),, u (4f) u (4g) u (4h) K, u (4) K, u (4), u (4k) uv, t, (4l),, u (4m) The nteger program presented n (4), s both accurate and complete. However, solvng t to exact optmalty s computatonally nfeasble when the nput sze s large. Ths s due to the nature of the channel assgnment problem, whch s nherently NP-Hard. If one relaxes the bnary varables x (u) and y (u) to freely take on fractonal values between 0 and, then the lnear nteger program n (4) degrades nto a lnear program wth contnuous varables only, and can be solved effcently usng nteror-pont algorthms or smplex algorthms. Snce the soluton to the LP relaxaton may assgn fractonal values to x (u) and y (u), t s therefore not feasble n general. Nonetheless, the maxmum throughput computed from the LP relaxaton provdes an upper-bound for the maxmum throughput from the orgnal nteger program. We next descrbe two algorthms for solvng the nteger program, and use the LP relaxaton as a benchmark for ther performance n smulaton studes later on. Note that f the throughput acheved by the algorthm s close to that acheved n the LP relaxaton, then t s even closer to the real feasble maxmum throughput (the optmal soluton to the nteger program). D. A greedy channel assgnment algorthm The frst of our two channel assgnment algorthms s relatvely straghtforward. The algorthm seeks to greedly assgn channels to create a vable, nterference-free route from the source to each and every recever. We begn by solvng the followng lnear program (see also [2]) that computes the optmal multcast flow usng network codng. f t( uv) Maxmze d (5) Subect To f t( ts) =d f t( vu) = 0 f t( uv) c (u) f t( uv) 0 t (5a) t, v (5b) t, u (5c) t, uv, u (5d) The constrant n (5c) captures the wreless broadcast property, and further bounds the total outgong flow from a node u to the total avalable capacty of all rados on u. The lnear program n (5) does not compute a meanngful flow snce channel assgnment and nterference effects are not taken nto account. Nonetheless, t does gve us an ndcaton of how mportant nodes are n the optmal flow. Whle a more fnegraned approach can be consdered by examnng the amount of outgong flow on each node, we choose nstead to consder all nodes wth non-zero flow as a frst approxmaton. Let these nodes be denoted by the set W. We can then safely prune all other nodes n the network and consder only nodes n W for channel assgnment. The greedy channel assgnment algorthm proceeds n a breadth-frst search fashon, begnnng at the source. If there s an avalable rado at a node, we assgn the frst avalable channel to that rado for transmsson. Channel avalablty depends on the nterference model we choose; n the case of the protocol nterference model, we choose the frst avalable channel that s not beng used for transmsson by nodes wthn 2 hops. Next, we assgn a free rado at each neghbour of the

6 Input: W, set of nodes to consder for assgnment NotDone := True ; whle NotDone do Vsted := ; enqueue(tovst, s); whle ToVst do u := dequeue(tovst) ; Vsted := Vsted + {u} ; foreach v N(u) v W do f v/ Vsted and v/ ToVst then enqueue(tovst, v) Pck avalable rado of wth hghest capacty ; Assgn frst avalable channel to ; f no avalable rado then f u = s then NotDone := false ; else contnue ; Algorthm : A greedy channel assgnment algorthm current node to lsten on the chosen channel. Ths establshes a connecton between the node and ts neghbours. We contnue to assgn channels n ths fashon (breadth-frst) untl each recever has been assgned a lstenng channel. At ths pont, we have a feasble channel assgnment to accommodate flow from the source to all recevers. However, the process has thus far only used one rado each for transmsson and recepton on every node. Snce nodes may have more than two rados, the breadth-frst assgnment process repeats agan begnnng at the source, whle ensurng that these new channel assgnments do not nterfere wth the exstng assgnment. Ths process repeats untl there are no longer any unassgned rados. Once ths channel assgnment stage s complete, the varables x (u) and y (u) n (4) take on fxed values, and we can then solve the resultng degraded lnear program of IP (4) for the optmal multcast flow. The complete greedy channel assgnment algorthm s shown n Algorthm. E. An teratve channel assgnment scheme The greedy channel assgnment scheme n Algorthm can be mproved. In ths secton, we pursue a soluton that teratvely mproves the exstng channel assgnment based on the computed flow graph. Our soluton s nspred by prmaldual algorthms, n partcular the scheme used by L [9]. The prmal-dual method of L seeks to maxmze multcast flow n undrected networks teratvely, where n each teraton, the computed flow s used to update the bandwdth allocaton n each orentaton of the undrected lnks n the network. Our teratve algorthm operates n a smlar manner, usng the computed flow to gude and mprove the exstng channel assgnment n each teraton. The mproved algorthm proceeds n two phases. In the frst phase, we run the greedy assgnment scheme teratvely on the resdual network resultng from the optmal flow computed by (4), and n the second phase, we make adustments to the assgnment obtaned n the frst phase to mprove channel and rado usage. Input: G =(V, E), graph model of wreless network Compute Maxflow, let W be set of nodes wth non-zero flow ; Run greedy assgnment scheme n Algorthm ; NotDone := True ; whle NotDone do Compute optmal flow wth (4), let f be flow graph ; Construct resdual network G usng f ; Compute Maxflow, let W be set of nodes wth non-zero flow ; Run greedy assgnment scheme n Algorthm ; f No Avalable Rado/Channel or Maxflow s 0 then NotDone := False ; Vsted := ; I := ; Enqueue(ToVst, s) ; whle ToVst do u := Dequeue(ToVst) ; f K, R(u) y (u) =then foreach v N(u) do f R(v) x (v) =then f f (v) > 0 and r R(u) yr(u) =0 and x r(u) =0, K then x r := ; else x (v) := 0 ; y (u) := 0 ; (w) := 0, w N(v) y y r (w) =; I := I + {u, v, w} ; foreach v N(u) do Enqueue(ToVst, v) ; NotDone:=True ; whle NotDone do Run Algorthm on all nodes excludng I ; Compute optmal flow usng (4) ; f No mprovement n flow rate then NotDone:= False ; Algorthm 2: An teratve channel assgnment scheme Smlar to Algorthm, we begn by computng the max flow and consderng only nodes wth non-zero flow for channel assgnment. Channels are assgned n a greedy fashon va a breadth-frst search begnnng at the source node as before. At ths pont, we have an ntal channel assgnment, whch we then use to solve the degraded lnear program of (4). Usng the flow graph obtaned from the degraded lnear program of (4), we construct the resdual network by removng rados wth non-zero flow from each node. Ths results n decreased capacty n each node, and once agan, we repeat the process of computng maxflow and greedy assgnment. We contnue n ths fashon untl we no longer have any free rados/channels or the max flow computaton yelds a zero flow rate. At the end of the frst phase, we may stll have neffcent channel assgnments. In the second phase, we examne the channel assgnment as well as the flow graph, and adust the assgnment to ether mprove bandwdth utlzaton or free channel usage for possble use n future assgnments.

7 Usng a breadth-frst search, we look for a node u that s lstenng to some channel, such that the ncomng flow on that channel s zero. Ths ndcates ether one of the followng two possbltes: u has been assgned to lsten to some neghbour v N(u) that s not forwardng flow on channel, or v s ndeed forwardng data but they are not ntended for u. In ether case, we can take remedal acton to mprove channel assgnment. If the outgong flow from v on channel s zero, then we can safely release the channel from use by u, v and all w N(v) that are also lstenng on. If v s transmttng a non-zero flow rate, then we assgn a new free rado of v to transmt on free channel (f both are avalable), and assgn u to lsten on nstead of. Due to the process of releasng channels that are not beng used, we now have more channels avalable, whch may be used by other nodes. We then teratvely do the followng; run the greedy assgnment scheme once agan whle excludng the nodes that released channels n the begnnng of the second phase, and compute the flow. If rados newly assgned wth the newly avalable channels have non-zero flow, we keep the assgnment; otherwse, we release t and repeat the assgnment phase whle excludng these nodes from assgnment as well. We contnue n ths manner untl the acheved flow can no longer be mproved. We state the entre procedure n Algorthm 2. For succnctness, we slghtly abuse notaton n Algorthm 2 and denote the outgong flow on node u on channel wth the varable f (u), whch has the followng nterpretaton f (u) = max t T f t ( uv) IV. SIMULATION RESULTS In order to study the effect of the number of rados and avalable channels on end-to-end multcast throughput, we ran smulatons on varous network topologes. We generated wreless networks usng the unt dsk graph model, where all nodes have a unform radus of communcaton, r. Usng a square grd of approprate sze, we generated nodes by assgnng them to cells n the grd at random. Lnks between nodes were created accordngly based on the parameter r. For a gven network, the choce of nodes n the multcast group was chosen at random. Each node s equpped wth multple rados, and the capactes of these rados are randomly dstrbuted between 0 and 50. As stated before, nteger programs are NP-hard and hence, solvng the nteger program n (4) for large networks s a computatonally ntensve process. Hence, for small networks, we solved the nteger program (IP) n (4), the lnear program (LP) relaxaton of (4) and the solutons based on Algorthm (Greedy) and Algorthm 2 (Iteratve), whle for larger networks, only the results of the last three are compared. The throughput shown for each soluton s the average computed from 4 separate smulaton runs over randomly generated topologes. To solve nteger and lnear programs, we employed the GNU Lnear Programmng Kt [3]. In partcular, the branch and cut 647!89!047:3;9<=3><8 &) %% %) $% $) #% #) "% ")!%!) %?@ A@ A80;,8-+0 B;007C )! " # $ % & ' ( *+,-.,/.023,440.5 Fg. 2. End-to-end throughput for networks of sze 0 wth a multcast group sze of 5. Each node s equpped wth 2 rados and the number of avalable channels s vared from 2 to 8 67-!8/!37-9:4/;<:2;8 *% )$ )% ($ (% '$ '% $$ $% #$ #% "$ "%!$!% &$ &% $ % =>?>?834,8.@3 A433-B! " # $ +,-./02345/-3 Fg. 3. End-to-end throughput as a functon of the number of rados n each node for networks of sze 0. The multcast group sze s 5, and a total of 6 channels are avalable for use method was used when solvng nteger programs whle the classc smplex algorthm of George Dantzg [32] was used to solve lnear programs. Fgure 2 shows the end-to-end throughput as computed by the IP, LP as well as the Greedy and Iteratve solutons of Algorthm and Algorthm 2 respectvely for networks of sze 0, wth a multcast group sze of 5 nodes, where each node s equpped wth two rados. We vared the number of avalable channels and studed the effect on the computed throughput. Wth fewer than 2 channels, no feasble multcast flow was found, snce wth a sngle channel, multcast would only be feasble f all recevers were neghbours of the source node. As we ncrease the number of channels, the upper bound as computed by the LP ncreases, but saturates at 5 channels onward. At ths pont, the number of rados per node s the lmtng factor and further mprovement s not possble unless more rados per node are provsoned. The optmal throughput as computed by the IP shows ncreased throughput as the number of channels s ncreased, and the true saturaton pont s acheved at 6 channels onward. The Greedy algorthm s only able to mprove throughput when the number of channels ncreases from 2 to 3, and s unable to utlze further addtonal channels. In contrast, the Iteratve assgnment scheme

8 647!89!047:3;9<=3><8 &) %% %) $% $) #% #) "% ")!%!) A80;,8-+0 B;007C )! " # $ % & ' ( *+,-.,/.023,440.5 <89!-/!,892=>0/?@>A?-!&"!%"!$"!#"!!"!"" *" )" (" '" &" %" $" #"!" " 2 BC D-,0E-4F, G0,,9H!" #" $" %" &" '" (" )" *"!"" +,-./02345,26728/9,:; 2 Fg. 4. End-to-end throughput for networks of sze 00 wth 0 multcast recevers and 2 rados per node, as a functon of the number of avalable channels Fg. 6. End-to-end throughput for varous network szes. Each node has 4 rados and there are 8 channels avalable for assgnment 67-!8/!37-9:4/;<:2;8 &!% &&$ &&% &%$ &%% *$ *% )$ )% ($ (% '$ '% $$ $% #$ #% "$ "%!$!% &$ &% $ % =>?834,8.@3 A433-B! " # $ +,-./02345/-3 Fg. 5. Effect of the number of rados per node on end-to-end throughput for networks of sze 00. The number of avalable channels s 8, and there are 0 multcast recevers acheves hgher throughput than the Greedy soluton from 4 channels onward, and contnues to ncrease throughput wth the ncreased avalablty of channels before saturatng at 6 channels onward. At ths pont, the Iteratve algorthm acheves a throughput that s close to 75% of the optmal. In Fgure 3, we study the effect of the number of rados per node on the acheved throughput, whle fxng the number of avalable channels at 6, on networks wth 0 nodes. Wth a sngle rado, acheved throughput s 0. Ths s because at least two rados are needed, one to transmt and one to lsten, as captured by the constrants n (4). As we ncrease the number of rados, the throughput computed by the IP ncreases. More rados lead to ncreased bandwdth per node as long as there are suffcent channels avalable for an nterference-free channel assgnment. At 2 and 3 rados per node, the Iteratve algorthm sgnfcantly outperforms the Greedy soluton. The performance of the Greedy algorthm only mproves when more rados are avalable. Ths ndcates that the Iteratve algorthm s more effcent at utlzng avalable resources when there s a scarcty of rados. In order to study the effect of avalable rados and channels n a larger network, we abandon the IP and focus on the endto-end throughput as computed by the LP and the proposed algorthms. The former provdes an upper bound on the optmal throughput, whle we wsh to study the performance of the latter n larger networks. Fgure 4 shows the throughput as computed by the LP and both algorthms for networks of 00 nodes and 0 multcast recevers, wth two rados per node. Smlar to Fgure 2, we observe an ncrease n the upper bound of the throughput as the number of channels ncreases up to a pont before saturaton, whch occurs at 4 channels onward. Once agan, we attrbute ths behavour to the lmted number of rados. The Greedy soluton s able to ncrease throughput margnally up to 6 channels before saturaton occurs. The Iteratve assgnment algorthm, on the other hand, performs sgnfcantly better than the Greedy algorthm, and s able to contnually mprove throughput, achevng close to 75% of the upper bound on end-to-end throughput when 8 channels are avalable. We also examned the effect of varyng the number of rados n the same network settng wth 00 nodes and 0 multcast recevers. Avalable channels were fxed at 6, and Fgure 5 shows the end-to-end throughput as computed by the LP and both algorthms. As the number of rados ncreases, the upper bound on achevable throughput grows. The Iteratve algorthm outperforms the Greedy soluton n all cases, and s able to ncrease throughput as the number of avalable rados ncreases. The Greedy soluton s also able to ncrease throughput wth more rados, but shows only margnal mprovement when the number of rados ncreases from 4 to 5. Agan, ths shows that the Iteratve algorthm computes channel assgnments that are more effcent at utlzng the avalable rados for hgher multcast throughput. Fnally, we study the effect of network sze on the performance of the LP and the algorthms, as shown n Fgure 6. The multcast group sze s 0, each node s equpped wth 4 rados and there are 8 avalable channels. Here, we see no notceable trend as the network sze ncreases. However, we note that the Iteratve algorthm sgnfcantly outperforms the Greedy soluton for all network szes.

9 V. FURTHER WORK Due to the NP-hard nature of the channel assgnment problem, t s nfeasble to compute the optmal multcast flow for very large mult-channel mult-rado networks. A lttle thought shows that the problem nherently les n the dscrete nature of orthogonal channels, wth non-overlappng spectrum. The latter s a concern, snce the IEEE 802. standard specfes 3 channels, of whch only 3 do not overlap. The above motvates a new possble soluton. Instead of consderng the predefned central frequences for channels, we allow rados to tune nto any frequency n some specfed spectrum range. We seek to answer the followng queston: what s the opportunty cost of a predefned spectrum to channel allocaton as opposed to a freely tunable rado n the spectrum? Specfcally, takng nto account nterference effects, can freely tunable rados acheve better throughput n mult-rado envronments? We have formulated a network model and a convex program to compute optmal multcast throughput n such a scenaro. Frst, we translate the network n the followng fashon; we create as many copes of each node as there are rados. If node u n the orgnal network has R rados, then n the new network we wll have nodes u... u R. Then, for each neghbour node v N(u) n the orgnal network, we add edges between every node u and v n the new network. Further, we add edges wth no capacty lmt between every node u and u to model the unrestrcted nternal flow from one rado to another n a sngle node. Let ths new network be the graph G = (V, E, E I ) where V and E are the sets of nodes and edges respectvely, whle E I s the set of nternal edges between nodes. Fgure 7 shows an example of the network translaton model. We now state the convex program to compute optmal throughput n ths network f t( uv) Maxmze d (6) Subect To f t( ts) =d f t( vu) = 0 f t( uv) f t(u) f t(u) c (u) t T (6a) v V, t T (6b) t T, u V (6c) t T, u V (6d) c (u) =f(α(u),α(v) v N I(v))c(u) u V (6e) K α(u) K 2 u V (6f) f t( uv),f t(u) 0 u V, uv E (6g) The convex program n (6) s smlar to the lnear program n (5), but dffers n that the capacty of a node s now a functon f, of the transmsson frequency chosen by the node, α(u), and the transmsson frequences chosen by the set of nodes wthn nterference range, N I (u). The functon f represents the amount of nterference caused by the choce of transmsson frequences of nodes, and has range between 0 and. Ko et al. [4] choose a convex functon that depends on the amount of channel overlap between adacent channels n ther work. We beleve a smlar dea would work here, but we were unable to run smulatons usng (6) due to u v w Fg. 7. An example of network translaton to an equvalent network n whch each node represents a rado. The lnks shown usng dotted lnes represent nternal, uncapactated lnks between rados wthn a sngle node n the orgnal network. The sold lnes show lnks between nodes n the new network that represent connectvty n the orgnal network tme constrants. Fnally, note also that K and K 2 n the convex program represent lower and upper bounds on the usable frequency spectrum. There s one more possblty to explore. One can consder the lnear programmng relaxaton of (4) and treat the fractonal values of x (u) and y (u) obtaned as an ndcator of how long rado should spend transmttng and recevng on channel respectvely. Ths rases a new algorthmc problem, gven fractonal values of x (u) and y (u), can we use ths to gude a schedulng algorthm that assgns channels for a fracton of a fxed tme slot? We beleve that usng ths dynamc channel assgnment method, we can acheve hgher throughput than the statc assgnment we have studed throughout ths paper. u u VI. CONCLUSION In ths paper, we study multcast usng network codng n wreless envronments n whch nodes are equpped wth multple rados, capable of usng channel dversty n the form of non-overlappng transmsson frequency bands. Prevous work on mult-channel networks have focused on uncast or mult-sesson uncast. In contrast, we focus on the optmal multcast problem. We have developed a model that s complete and optmal. Moreover, our model explots the broadcast property n wreless networks, by employng network coded transmssons to effcently utlze avalable bandwdth. The channel assgnment problem, smlar to colourng problems n graphs, s NP-hard, whch s reflected n the nteger program model derved. We have presented a smple, greedy channel assgnment algorthm as well as an mproved soluton that performs channel assgnment teratvely usng the resdual network of the flow computed. The results from smulatons performed have provded us wth an nsght nto how end-to-end throughput s affected by the avalablty of both channels and rados per node. We fnd that the teratve algorthm utlzes the avalable rados and channels more effcently and thereby acheves hgher end-toend throughput as compared to the greedy channel assgnment scheme. We ntend to pursue the work presented here further wth the ultmate goals of characterzng the feasble flow regon of mult-channel mult-rado wreless networks, and 2 v v 2 w w 2

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