The Effect of Multi-Radio Nodes on Network Connectivity A Graph Theoretic Analysis

Transcription

1 The Effect of Multi-Radio Nodes o Network Coectivity A Graph Theoretic Aalysis Adrás Faragó Departmet of Computer Sciece The Uiversity of Texas at Dallas Richardso, Texas, U.S.A. farago@utdallas.edu Stefao Basagi Departmet of Electrical ad Computer Egieerig Northeaster Uiversity Bosto, MA, U.S.A. basagi@ece.eu.edu Abstract We aalyze the gai i etwork coectivity that is obtaied by implemetig multiple radio iterfaces i the odes. The multi-radio odes ca act as if the etwork had effectively multiple physical layers. We model such a etwork topology by a multigraph ad capture the gai by itroducig the ovel graph theoretic cocept of the multigraph advatage. Whe applied to coectivity, it is the surplus of coectivity over the sum of the idividual coectivities, as we put together several graphs to form a multigraph sum. We prove i a radom graph model that this results i a strict super-additive behavior, always yieldig multigraph advatage. Moreover, for the most importat desity rage, called moderately dese regime, we prove that the gai grows to ifiity with the graph size ad the percetage (relative) gai remais costat ad does ot vaish with growig etwork size. 1 I. INTRODUCTION AND MOTIVATION The growig importace of wireless etworks is a major ad lastig tred i the etworkig ladscape. It also comes had i had, however, with the icreasig diversity of wireless etworkig solutios ad stadards. Some of them are already widely used, such as variats of IEEE wireless LANs (802.11a, b, g), others are emergig, such as IEEE persoal area etworks, Zigbee, Bluetooth, broadbad wireless (IEEE ), a variety of sesor etworkig solutios, etc. Aother well visible tred is that radio iterfaces are rapidly gettig more ad more iexpesive ad physically small. It is ow techically ad ecoomically quite feasible to equip wireless etwork odes with several radio trasmitters/receivers. This creates a eviromet where the etwork effectively has multiple physical layers. For example, it is already quite commo that a laptop or a PDA has both a IEEE card ad a Bluetooth iterface. Give the tedecy of decreasig price ad shrikig physical size, it is quite likely that the tred of havig multiple radio iterfaces will get eve more prevalet i the future. Thus, may of the ubiquitous wireless etwork odes will likely be capable of operatig with multiple physical layers. A few examples, preset ad future: Multi-radio odes (e.g., laptops with multiple radio iterfaces, as metioed above). 1 Research is supported i part by NSF Grats CCF , Multi-chael radio eviromets that logically act as multiple radios. Note that the IEEE stadard defies multiple chaels that are oly partly utilized today. Multiple atea systems that implemet several idepedet chaels via sophisticated physical layer techiques, such as beamformig, Space Divisio Multiple Access (SDMA), cooperative codig, multiple-iput/multipleoutput (MIMO) systems, etc. Combiatio of radio ad ifrared iterfaces i a ode. Utilizatio of low-power wireless techologies, such as RFID solutios with eergy harvestig, various wireless sesor platforms, etc. Combiatios of radio ad other potetially possible wireless trasmissio techologies, such as free space optical trasmissio, laser beams, etc. Thus, while the techical possibility of multiple physical layers is already quite clear today, it is much less obvious how ca it be efficietly utilized to gai sigificat improvemet i the overall etwork performace. Or, from the practical/ecoomical poit of view, the ultimate questio is: Will the multi-radio etwork developmet lead to sufficiet performace improvemet that justifies the ivestmet? We believe that to aswer this questio it is ecessary to develop methods that ca somehow quatify the etwork performace gai that is obtaied via multiple physical layers. The modelig approach should be sufficietly geeral, too, so that it ca capture a variety of importat scearios. To develop such a methodology is the core motivatio of this paper. Before discussig our modelig approach, we briefly review the previous work related to multi-radio etworks i the ext sectio. We ote i advace that our pla is to pursue a somewhat more abstract approach tha what is typically foud i the etworkig literature, sice we seek a aveue of modelig that is geeral eough to icorporate may particular scearios. II. PREVIOUS WORK ON MULTI-CHANNEL/MULTI-RADIO NETWORKS The availability of multiple radio iterfaces ad multiple chael per radio i multi hop etworks have raised sigificat

2 iterest i the academic ad idustrial commuity due to the desig ad deploymet of mesh wireless etworks, i.e., multihop etworks where some of the odes, usually less mobile, provide backboe coectivity to mobile ed users [2], [10]. As a cosequece, whether related directly to mesh etworkig or just cosiderig geeric multi-radio systems, a good umber of papers have recetly appeared at premier etworkig cofereces i this area. Although quite differet i ature, all works o multi-radio/multi-chael systems strogly suggest ad demostrate the advatage of exploitig the radio diversity provided by multi-chael/multi-radio systems, ad i geeral, of havig multiple radio iterfaces for cosiderable performace improvemet. Various solutios have bee proposed for sigle-radio, multi-rate systems. Some techologies, such as IEEE a/b/g have multiple trasmissio rates, so that they ca accommodate multiple chael coditios, ehace throughput ad decrease iterferece losses. For example, i the work by Awerbuch et al. [4] it is show that the miimum-hop routig strategy, which is usually deployed i sigle rate etworks, is sub-optimal i multi-rate etworks. Therefore, the authors propose a differet lik selectio metric (Medium Time Metric, MTM) that favors shorter, higher throughput ad more reliable liks istead of the loger liks that regular routig would select. Iterferece-aware topology cotrol ad routig is discussed i [23] where sigle-radio (IEEE b) odes are cosidered. By takig advatage of the multiple chaels of the radio, the authors defie a way of choosig chaels so that the resultig topology is iterferece-miimum amog all possible k-coected topologies. For splittable traffic demads, the authors the preset the Badwidth-Aware Routig (BAR) protocol, which is demostrated to outperform routig o a sigle-chael topology. Chadra et al. [7] advocates the use of sigle-radio systems i a multi-lan eviromet. A wireless device could coect to multiple wireless LANs simultaeously despite havig a sigle radio card. The paper describes a software layer approach to this problem, called MultiNet, that virtualizes a sigle wireless card ad facilitates parallel multiple coectio. The capacity of wireless multi-radio, multi chael etworks have bee addressed, e.g., by Kyasaur ad Vaidya i [16]. I etworks with static odes, each with m radios ad c chaels, 1 m c, this work shows that the capacity of the etwork exhibits bouds that are differet from those established by the paper o sigle-radio/sigle-chael systems by Gupta ad Kumar [11]. I particular, the multiradio, multi chael bouds deped o the ratio betwee c ad m. A characterizatio of the achievable rates i a mesh etwork with orthogoal chaels is give i [14] where the authors determie ecessary ad sufficiet coditios for achievig a rate vector i systems where the chaels do ot iterfere with each other. By extedig a former solutio of theirs [15], the authors derive joit routig ad schedulig algorithms for systems that are full duplex ad equipped with multiple radios. Routig for a multi-radio etwork with static odes (like a wireless commuity etwork [24]) has bee preseted i [8]. The authors propose a ew metric for routig i etworks with multi-radio odes, takig ito accout loss rate ad lik badwidth ad defie a correspodig path metric that also cosiders iterferece amog liks usig the same chael. The performace of the ew metric is demostrated via a 23 ode testbed with odes moutig two IEEE a wireless cards. The same testbed is also described i [5] where geeral desig guidelies are provided for buildig multi-radio systems. This work also revisits the stadard problems i wireless etworkig i light of havig odes with multiple radio cards. The advatage of usig multi-radio odes has also bee demostrated i [17], where a sigle-hop wireless system is preseted, termed Multi-Radio Diversity (MDR), which uses path diversity to improve loss resiliecy. The paper demostrates the multi-radio advatage by showig a throughput gais up to 2.3 over sigle radio commuicatio schemes (their testbed is based o odes supportig multiple IEEE a wireless cards). Shi et al. [20], [21] advocate the use of multi-radio to icrease the capacity ad performace of cellular etworks. The idea i this case is to exploit the best available lik to the base statio. The authors evisio a eviromet i which relay etworks are dyamically formed so that wheever o acceptable direct liks are available from a mobile ode to the base statio, a multi-hop path ca be foud. Algorithms for efficiet formatio of multi-hop relay etworks are preseted ad evaluated for latecy, sigal overhead, gateway load (etwork formatio) as well as path legth ad lik sharig (relay etwork fuctios). Chael assigmet ad routig for multi-hop wireless mesh etworks is formulated mathematically as a joit optimizatio problem, by Alicherry et al. i [3]. The authors cosider iterferece costraits, the umber of chaels per radio, ad the umber of radios per ode. By solvig the correspodig mixed iteger liear programmig model a cetralized algorithm is proposed that optimizes the overall etwork throughput subject to fairess costraits. Distributed chael assigmet i multi-hop, multi-radio etworks has bee explored i [22]. The problem is show to be NPhard eve if the paths betwee ay two odes are give. A radomized assigmet scheme is the proposed (Skeleto Assisted partitio FrEe, or SAFE) that maitais etwork coectivity. A method for iterferece-aware cael assigmet has bee proposed by Ramachadra et al. [19]. By modelig iterfereces betwee each mesh multi-radio router via a ewly defied multi-radio coflict graph, a ew chael assigmet algorithm is desiged that addresses explicitly the iterferece problem. Simulatios ad testig o a IEEE testbed are performed that yield performace gais i excess of 40% with respect to static (o-iterferece aware) chael assigmet solutios. I order to ease the applicatio access to multi-radio cards o a sigle ode the Multi-radio Uificatio Protocol (MUP) has bee proposed by Adya et al. [1]. The goal of MUP is

3 coordiatig the operatios of the differet (IEEE ) cards tued to o-overlappig frequecy chaels by optimizig local spectrum usage. The advatage of usig this kid of software iterface has bee demostrated via s2- based simulatios. Results show that TCP throughput ad user perceived latecy sigificatly improve uder dyamic traffic patters over realistic topologies. (a) G 1 : λ(g 1 )=1 III. MODELING APPROACH FOR NETWORK TOPOLOGY ANALYSIS Our fudametal model of the etwork topology for a etwork with multiple physical layers is a multigraph, possibly with labeled edges. As it is well kow, a multigraph differs from a ordiary graph i that multiple edges ca coect ay pair of vertices. Naturally, the vertices represet etwork odes, while the edges represet the liks i the various physical layers, geerated by the differet radio iterfaces. The labels o the edges distiguish their affiliatio to the various layers. As graphs have log bee used to model etwork topologies, oe may rightfully ask the questio at this poit: Ca a multigraph lead to ay essetial ew isight? I what follows we show that this model yields iterestig ad otrivial ovel problems. A. The Multigraph Advatage Cosider a wireless multi-hop etwork with omidirectioal ateas ad let its etwork topology be represeted by the graph G 1 i Figure 1(a). Assume that we equip the same odes with aother physical layer, such as a secod radio with beamformig capabilities. This secod physical layer geerates aother etwork topology, show by graph G 2 i Figure 1(b). If we put them together, we obtai the topology of the combied system, show by the multigraph G i Figure 1(c). Let us call the above mergig operatio of graphs the multigraph sum of G 1 ad G 2. The operatio will be deoted by. Thus, we obtai G as G = G 1 G 2. The operatio ca be exteded to more compoets i a atural way, so we ca take the multigraph sum of ay umber of graphs or multigraphs: N i=1 G i = G 1 G 2... G N. Note that by assigig labels to the edges, the multigraph sum becomes ivertible, that is, we ca recover each idividual G i from N i=1 G i. Havig itroduced the multigraph sum, let us ow cosider a curious property of it, which we call multigraph advatage. We explai it through a importat graph parameter, the edgecoectivity (for short, we simply call it coectivity), deoted by λ(g). This is the miimum umber of edges that ca discoect the graph, whe these edges are deleted. I other words, this is the size of a miimum cut i the graph. I case G is a multigraph, the parallel edges are all couted Figure 1. (b) G 2 : λ(g 2 )=2 (c) G = G 1 G 2 : λ(g) =4 The multigraph advatage: Edge coectivity whe we cosider the size of a cut. The coectivity provides a importat characterizatio of the etwork topology. For example, it tells how vulerable is the etwork to lik failures. It also shows how rich is the etwork i lik-disjoit routes. 2 Now oe ca easily see that λ(g 1 G 2 ) λ(g 1 )+λ(g 2 ) (1) always holds. More geerally, we have ( N ) N λ G i λ(g i ). (2) i=1 The reaso is that the size of ay give cut i the multigraph sum is just the sum of the sizes of the correspodig cuts i the compoets. Thus, the coectivity of the multigraph sum caot be smaller tha the sum of the coectivities of the compoets. I fact, from the above reasoig oe might first expect that actually the iequalities (1) ad (2) always hold with equality. This is, however, ot the case. Let us look at the graphs i Figure 1. Oe ca directly check that λ(g 1 )=1 (Figure 1(a)) ad λ(g 2 )=2(Figure 1(b)). O the other had, as show i Figure 1(c), λ(g) =4, which is strictly greater tha the sum. The above small example shows what we call multigraph advatage. Regardig coectivity, it meas that the coectivity of the multigraph sum ca be strictly larger tha the sum of the coectivities of the compoets. Thus, the i=1 2 This is because it guaratees that betwee ay pair of odes there are at least λ(g) disjoit routes. (This fact is a cosequece of Meger s Theorem i graph theory.)

4 etwork topology has a quatifiable extra beefit from takig the multigraph sum, represetig multiple physical layers. Oe may reasoably ask at this poit: Is this behavior just the cosequece of specially chose examples or is it somehow typical? I the followig sectios we show that it is quite typical. IV. MULTIGRAPH ADVANTAGE IN A RANDOM NETWORK MODEL Mobile wireless etworks are ofte modeled by various types of radom graphs. I what follows, we first discuss the choice of the cosidered radom graph model, the we formally state the thorems showig that it exhibits sigificat multigraph advatage. Ufortuately, size limitatios do ot allow us to iclude the (legthy) formal proofs, they will be preseted i the joural versio of the paper. A. The Choice of the Radom Graph Model The type of radom graph that is most frequetly used to model mobile wireless etworks is the radom geometric graph [18]. I the simplest case it is geerated by radomly placig poits i a plaar domai, typically a regular oe, such as a square. The poits represet the radomly positioed etwork odes. Ay two poits are coected with a edge (represetig a etwork lik), wheever their distace is at most a give value r, that stads for the trasmissio radius. There are various geeralizatios of the above simple model. For example, the domai ca be irregular ad it may be part of a higher dimesioal space, possibly with a differet metric. To our kowledge, the most geeral versio that still has a geometric flavor is based o pre-metric spaces [9] subsumig all previously studied geometric radom graph variats. O the other had, the oldest ad most studied radom graph model is the oe i which all edges are chose idepedetly with the same probability [6]. This model is referred to as idepedet-edge radom graph, or Erdős-Réyi radom graph, or sometimes Beroulli radom graph. While this model lacks the edge correlatios iduced by geometry, evertheless it has a umber of importat advatages: I situatios where the trasmissio radius is comparable with the domai diameter, the mai reaso for a missig lik is ot the distace, it is the presece of radom obstacles or radom variatios i the radio propagatio. I such a case a idepedet edge model ca be adequate, or eve more accurate tha a geometric radom graph. Some studies have foud (see [12], [13]) that if the radio propagatio model is more realistic, i.e., it takes ito accout statistical variatios aroud the mea power, the it teds to decrease lik correlatios ad the radom graph becomes similar to the idepedet-edge model. If the etwork applies power cotrol, it teds to couterbalace the effect of distace, especially if the distaces are ot too large. This agai poits to the applicability of the idepedet edge model. The idepedet-edge model is much more ameable to mathematical aalysis. The above reasos justify that we choose the idepedetedge model for our ivestigatios. B. Results We cosider the followig situatio. Let G,p deote a radom graph o odes, with edge probability p. Note that typically p is a fuctio of, i.e., p = p(), ad we are iterested i asymptotic properties, whe. A well kow property (see [6]) is that G,p is coected asymptotically with probability 1 if ad oly if log + ω() p() = where ω() is ay fuctio that teds to ifiity with. Iformally, this meas that the edge probability of log is the miimum requiremet for coectivity, as ω() ca ted to ifiity arbitrarily slowly. First we ivestigate the multigraph advatage i a situatio whe the etwork is moderately dese i the sese that the average degree is oly costat times larger tha the miimum eeded for coectivity. Defiitio 1 The radom graph G,p is said to be i the moderately dese regime if there exists a costat c>1, such that p() = c log holds. Now we cosider the followig situatio. Let G,p1 ad G,p2 be two idepedetly draw radom graphs o the same set of odes, with edge probabilities p 1,p 2, respectively. Assume that they are both i the moderately dese regime. Let us compare the coectivities λ(g,p1 ), λ(g,p2 ) ad the multigraph coectivity λ(g,p1 G,p2 ). We ca prove the followig result about the multigraph advatage regardig coectivity. Theorem 1 Let G,p1 ad G,p2 be idepedetly draw radom graphs i the moderately dese regime, o the same set of odes. Let their edge probabilities be p 1 () = a log ad p 2 () = b log with costats a, b > 1. The there exists a costat c = c(a, b) > 1, such that the asymptotic multigraph advatage regardig coectivity is at least c log. That is, λ(g,p1 G,p2 ) λ(g,p1 )+λ(g,p2 )+c log holds asymptotically with probability 1. Thus, as show by Theorem 1, the multigraph advatage is quite sigificat i the moderately dese regime, for two reasos. First, it teds to ifiity as the graphs grow. Secod, sice it is kow [6] that i the moderately dese regime the coectivity of the radom graph is O(log ), therefore, the

5 gai i coectivity is of the same order of magitude as the coectivity of the compoet graphs itself. I other words, there is a guarateed costat percetage of relative gai, ad this percetage does ot vaish as. The actual value of the costat c, as a fuctio of a ad b, ca also be computed, although somewhat idirectly, as follows. Let f(α) be a fuctio defied for α>1 by f(α) =αη α where η α is the uique solutio i ( 1, 0) of the equatio 1+αx = α(1 + x) log(1 + x). With this, c ca be expressed as c = f(a + b) f(a) f(b). Oe ca also prove that the fuctio f is strictly superadditive, so we get c>0. If the desity of the radom graphs are ot restricted to the moderately dese regime, the we caot guaratee a costat percetage gai i coectivity. Nevertheless, we ca still prove that there is a ozero multigraph advatage. This is made precise i the followig theorem. Theorem 2 Let G,p1 ad G,p2 be idepedetly draw radom graphs that are coected asymptotically with probability 1, but otherwise their desity ca be arbitrary. The λ(g,p1 G,p2 ) >λ(g,p1 )+λ(g,p2 ) holds asymptotically with probability 1. The message of Theorem 2 is that wheever the radom graphs have eough desity to be coected, the the multigraph advatage is asymptotically always preset. V. CONCLUSION We have aalyzed the effect of multiple radio iterfaces o etwork coectivity. This was captured by the ovel graph theoretic cocept of the multigraph advatage. We have proved i a radom graph model that this multigraph advatage is always preset, wheever the compoet graphs are coected. Moreover, for the most importat desity rage, called moderately dese regime, we proved that the gai grows to ifiity with the graph size ad the percetage (relative) gai remais costat ad does ot vaish with growig etwork size. The proofs ad supportig simulatios will be preseted i the joural versio of the paper. 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