Totally Disjoint Multipath Routing in Multihop Wireless Networks

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1 Totally isjoint Multipath Routing in Multihop Wieless Netwoks Sonia Wahate and Raouf outaba Univesity of Wateloo School of ompute Science Wateloo, anada bstact Speading taffic ove multiple paths has been shown to enhance netwok pefomance compaed to single path outing. Howeve, the oute coupling effect specific to wieless envionments (consequent to the shaed tansmission medium) can significantly educe the benefits of such an appoach. In this pape, we focus on the -path outing poblem in a non-mobile and non enegy-constained netwok. We evaluate the netwok pefomance and the effect of intefeence in a single soucedestination pai scenaio and fo multiple souce-destinations pais. In the fome case, we povide an analytical evaluation of the thoughput and popose a position-based algoithm to iteatively find a path between a souce and a destination. In the latte case, we demonstate how intefeence can significantly degade the nominal netwok capacity and void the benefits of multipath outing. I. INTROUTION Multipath outing is pomoted as an altenative to single path outing fo its potential to addess issues such as oute failue and ecovey, and netwok congestion. In enegy constained netwoks such as ad hoc netwoks and senso netwoks, it has been demonstated that multipath outing achieves highe pefomance than single path outing esulting in an oveall lowe powe consumption [] [11]. Recently, poposals fo multipath outing in wieless netwoks have flouished, the main contibutions focusing on eithe impoving path eliability though the establishment of backup paths [8] [1] [1] o on optimizing netwok esouce utilization via load distibution among seveal paths [9] [14]. Vaious path selection stategies have been poposed with diffeences in the paametes to optimize (packet dop atio, ovehead, o end-to-end delays), the level of fault toleance equied, o the congestion avoidance stategy (link-disjoint vs. node-disjoint paths). Seveal multipath outing appoaches consisted in adapting pominent single path outing potocols such as OV and SR [9] [13] fo multiple paths outing, by modifying the fowading pocess of esouce equests and the infomation maintenance at intemediay nodes [1] [1] []. These woks have consideed some (o all) of the constaints petaining to ad hoc netwoks: node mobility, enegy limita- This eseach is patially suppoted by NORTEL, ommunications and Infomation Technology Ontaio (ITO) and the Natual Sciences and Engineeing Reseach ouncil of anada (NSER). tions, high netwok density and limited computation capabilities. Howeve, only a few woks have actually consideed the effect of intefeence in multipath outing and, to the best of ou knowledge, none has povided an analytical evaluation of the netwok pefomance in this context. In this wok, we povide an evaluation of the thoughput in a multipath outing stategy while taking into consideation the impact of intefeence. We focus on netwoks with fixed and non-enegy constained wieless backbone, with potentially enhanced capabilities such as GPS systems. We adopt an incemental appoach to addess this poblem by fist consideing only the intefeence between a same souce-destination pai and next, between multiple souces and destinations. The emainde of the pape is oganized as follows. In Section II, we descibe the challenges of multipath outing in wieless envionments. We pesent the -path outing poblem in Section III. In Section IV, we povide an analytical evaluation of the netwok thoughput fo the single souce-destination scenaio and focus on the multiple souce-destination pais case in Section V. Section VI concludes this pape. II. MULTIPTH ROUTING IN MULTIHOP WIRELESS. Route coupling NETWORKS In wieless communications, sevee pefomance degadation can esult fom intefeence of concuent data tansmissions. The shaed tansmission medium constains all nodes in the intefeence ange of a sende o eceive to inactivity until completion of the ongoing communication. When adopting a multipath stategy, this poblem is futhe exacebated as, in ode to incease the nominal thoughput beyond what single path outing would offe, it is necessay to guaantee that the chosen paths do not intefee with each othe. The set of paths between a souce node s and a destination node t can be of two sots: link-disjoint paths: one communication link can not be shaed by seveal paths. node-disjoint paths: a same node can not be pat of seveal paths. To account fo the potential intefeence between concuent data tansmissions, it is impotant to distinguish the two following subcases, in which distinct paths can be:

2 1) edge-connected and theefoe intefee. This phenomenon of intefeence between two (o moe) paths is known as oute coupling. ) -edge-connected and do not intefee. This latte case is efeed to as totally disjoint paths in the emainde of this pape. To evaluate the impact of such topologies on the maximal achievable thoughput, let us conside that a souce node and a destination node ae hops away, with a nominal netwok capacity of (thoughput that can be achieved at the M laye in one hop [5]). In the example consideed, when the paths ae totally disjoint, the thoughput is limited to /5 (Fig. 1a) as the entie path is in the intefeence ange of the middle link (epesented in bold in Fig. 1). When the paths ae node-disjoint but ae 1-edge connected, the maximal thoughput is bounded by / (Fig. 1b). This thoughput is futhe educed to /9 when a node is shaed by paths (Fig. 1c).. Related Woks In [1], the authos emphasized the impact of oute coupling effect, which occus when two paths ae located close enough to intefee. They poposed a famewok fo multipath outing in ad hoc netwoks using diectional antennas. ontol packets ae exchanged in omni-diectional mode wheeas data packets ae tansmitted in diectional mode, theefoe allowing multiple tansmissions at the same time. This mechanism appeas nonetheless insufficient to pevent data collision as contol packets ae boadcast and can conceivably intefee with any ongoing data communication occuing in the vicinity of the sende o the eceive. Theefoe the usefulness of diectional antennas in this context is disputable. Inte-path intefeence has also been consideed in [19], in which the authos poposed a path selection mechanism based on the intefeence level between two paths (fo a given soucedestination pai), detemined empiically via measuements. Few woks have tied to deive an analytical fomulation of the pefomance of multipath and single path outing algoithms in multi-hop wieless netwoks. In [15], the authos assumed that, when the netwok density is high enough, outes can be appoximated by staight lines. Theefoe, by computing the numbe of outes going though a paticula node, the elayed taffic can be evaluated and the numbe of queued packets can be computed. simila appoach is used fo multipath outing but without specifically consideing the numbe of outes pe connection. [3] impoved ove the pevious wok by alleviating the esults obtained by [15] fom the assumption of unifom load distibution. The authos consideed that in k-shotest paths outing the outes between a souce and a destination ae contained within a ectangula aea and, given a destination node and a elay node, the position of the possible souce nodes can be detemined. ll the nodes satisfying these conditions ae assumed to be pat of a path going fom the souce to the destination. ut these two woks did not take into consideation the poblem of intefeence, limiting thei applicability to wieless netwoks. III. TOTLLY-ISJOINT -SHORTEST ROUTING. Poblem efinition In this pape, we focus on a special case of multipath outing: -path outing. efinition Given a diected gaph G(V,E) and two nodes (s, t) V, -path consists of finding two paths P 1 and P between s and t such that: a) all the nodes in P 1 and all the nodes in P fom a connected gaph; and b) thee exists no edge between any node in P 1 and any node in P.. ssumptions and esign ecisions In the emainde of the pape, the following assumptions and design decisions have been made. Fist, the nodes ae unifomly distibuted with a density ρ in an aea of adius R. Fo simplification, the intefeence ange is assumed to be the same as the tansmission ange (this does not impact the computation but simplifies the poblem fomulation). We futhe conside that the netwok density is high enough to appoximate the shotest path between any soucedestination pai by a staight line. Theefoe, the -path poblem can be educed to abstacting a path as a tajectoy and to detemining the pobability that a node is located on it. s we focus on totally-disjoint -shotest paths, the solution paths P 1 and P should be spaced by at least a distance of (intefeence ange) to pevent the oute coupling effect as peviously descibed. s we ae looking fo the shotest paths, the sets of possible solutions ae theefoe located in the bands of width (whose computation is futhe descibed). IV. THROUGHPUT ESTIMTION: THE SINGLE SOURE-ESTINTION SENRIO. Multipath outing vs. Single path outing To estimate the benefits of multipath outing ove single path outing, we un seveal simulations in the ns-.8 netwok simulato [4]. IEEE 8.11b is used as the undelying M potocol and UP as the tanspot potocol. The outing is static and no ovehead due to oute establishment is accounted fo. The netwok is composed of a souce node and a destination node located hops apat (Fig. ). We evaluate the thoughput fo a single path stategy, a multipath stategy with totally disjoint paths and a multipath stategy with intefeing paths (the paths ae 1m apat). single flow is sent fom Node to Node. The simulation paametes ae summaized in Table I. Souce Fig estination Netwok topology fo the multipath scenaio

3 Souce estination Souce estination Souce estination (a) without connection (b) with link connection (c) with node connection Fig. 1. ollision domains (shadowed aeas) (a) Single Path (b) Intefeing -path (c) Non-intefeing -path Fig. 3. Thoughput ataate Packet size Tansmission ange Taffic Tanspot Potocol Routing Potocol 11Mbps 15 bytes 5m R at.5 packets/s UP Static TLE I SIMULTION PRMETERS Fig. 3 epesents the thoughput measued at the destination node. We can obseve that the highest pefomance is achieved in the multipath scenaio without intefeence (Fig. 3c), wheeas the pefomance significantly deceases when intefeence is intoduced and with single path outing. This esult cooboates the necessity of using non intefeing paths when a multipath outing stategy is adopted.. Thoughput Estimation: Methodology lthough we ae focusing in this pape on the evaluation of the thoughput in a totally-disjoint -shotest path poblem, a simila method can be applied to addess the k-shotest paths outing poblem. Let and be a souce-destination pai and F a node pat of a path between and. The evaluation of the thoughput at F necessitates to evaluate the taffic geneated by the node itself and the taffic fowaded by F but geneated by nodes geogaphically distant fom F. Theefoe, the thoughput computation can be boken into two steps: Step 1: etemine the maximum numbe of paths going though node F. Step : etemine the pobability of paticipation of Node F in the fowading pocess. Indeed, being geogaphically located on the tajectoy between and does not necessaily imply a node paticipation in the fowading pocess. Fo instance, if the communication between node and node necessitates 4 hops, only 3 elay nodes ae needed, although moe nodes (depending on the netwok density) can be potential candidates.. etailed nalysis 1) Relay Taffic: To compute the numbe of paths going though a paticula node F, we need to locate the souce and destination nodes that can potentially have a path going though F. The method consists in computing the tangents to the cicle centeed at F of adius /. The tangents and thei paallel located at define an aea in which the souce and destination node should be located. The equation of the cicle F centeed at F with adius / can be staightfowadly deived as: x + y xx F yy F + x F + y F (/) = (1) The equation of the tangent to F at P 1 (x 1,y 1 ) is: xx 1 +yy 1 x F (x+x 1 ) y F (y +y 1 )+x F +y F (/) = () The coodinates of the intesection points (x,y ) and (x,y ) between the tangent and the cicle O centeed at can be detemined fom Eq. 1 and Eq.. Fo each point P 1

4 F, we can theefoe compute the aea whee the possible souce and destination nodes ae located, and by consequence the maximal numbe of paths going though F at P 1.We obtain: N max( F,P 1) = (x x 1 ) +(y y 1 ) (x x 1 ) +(y y 1 ) ρ (3) Fom this, we need to emove the paths whose length is less than (meaning that the souce node and destination node ae in diect tansmission ange of each othe). This can be deived by applying ofton s fomula [18]. Let us conside n points ξ 1,..,ξ n andomly distibuted on a domain S, let H be some event dependent on the nodes position. Let δs be a small pat of S. ofton s fomula states that: δp[h] =n(p [H ξ 1 δs] P [H])S 1 δs (4) We can theefoe obtain: P (X >)= 1 P (X >)= 1 x dx The total numbe of paths to be emoved is theefoe: N em = Let N F (α 1 ) be the numbe of paths going though F at P 1. N F (α 1 ) is the esult of subtacting Eq. 5 to Eq. 3. The next step of the computation consists in detemining the total numbe of paths fo all the points located on F.We fist pefom a tansfomation of P 1 s euclidian coodinates into pola coodinates: x1 = F cosθ F + cosα y 1 = F sinθ F + sinα The maximal numbe of paths N paths going though F can theefoe be obtained by summing up all the possible soucedestination pais on the tangents to F. N paths (, θ) = π (5) N F (α)dα () This can be evaluated with analytical methods. ) Expected Numbe of Fowading Nodes: Let us define the expected pogess as the distance coveed in 1 hop []. This paamete is of significant inteest in ou computation as it is diectly elated to the numbe of hops along a path fom a souce to a destination. The geate the expected pogess, the smalle the numbe of hops. This paamete depends on the netwok density and node distibution. Let z be the maximum expected pogess. With a unifom nodes distibution, the numbe of nodes follow a Poisson distibution. To have a maximum expected pogess z, thee should be at least one node in the aea located between and z, that is to say the pobability p that thee is no node between and z should be vey small: P (N = ) = e ρz = p We can thus deive the following equation: = log(p ) ρz The expected elay taffic pe node (, θ) is theefoe:. Validations (, θ) = () 1 ρz N paths(, θ) (8) To validate ou analysis, we used two methods: 1) We implemented a outing algoithm based on node positioning ) We computed the total elay taffic in a single path outing stategy and compaed it against the total elay taffic in a -path outing stategy. 1) Iteative Position-based Multipath Routing lgoithm: Given the assumptions of ou wok (fixed wieless backbone and possibility to easily obtain the nodes positions), we popose a localization-based outing potocol. The algoithm is implemented as follows. Let V be the set of nodes, S the souce node, T the destination node, N c the cuent elay node, N the next hop node and N oth the othogonal pojection of N on (S, T ). The algoithm consists in iteatively finding the next hop node on each path within the tansmission ange of the cuent elay node and satisfying the intefeence constaints (i.e. the chosen node should be in one of the bands descibed in Fig.??). lgoithm 1 Multipath Routing lgoithm N c = S set = V cuent distance = dist(s, T ) while set = do if dist(n,n oth ) > (/) && dist(n,n oth ) < (/)+ && dist(n c,n) <then if dist(n,t) <cuentdistance then N c = N cuent distance = dist(n,t) end if end if set = set{n} pick N in set end while The algoithm is un fo each souce-destination pai.

5 ) Single path outing: In ode to detemine the numbe of nodes involved in the total elay taffic fo a single path stategy, we need to compute: 1) the total numbe of paths ) the pobability that a path length is less than the tansmission distance (i.e. the souce and destination can communicate diectly) in ode to exclude these paths fom the set of feasible paths 3) the aveage path length M(R) between two nodes located in a disk of adius R given that the path length exceeds the tansmission ange The total numbe of paths N paths is staightfowadly obtained: N paths = ρπr (ρπr 1) (9) The pobability density fo the distance between two andom points located in a cicle of adius R can be expessed as [] [18]: p(x) = x R ( π accos( x R x πr (1 x ))) (1) 4R Theefoe, the pobability that the distance between nodes exceeds the tansmission ange can be deived easily: P (x >)= R p(x)dx (11) Finally, we need to calculate the mean distance between two nodes andomly dopped in a disk given that the distance between these two nodes is geate than a distance (tansmission ange). Let be the mean distance between a node located on the cicumfeence of a cicle of adius R and any othe node located in the cicle whose distance exceeds the tansmission adius. can be expessed as follows: = 1 πr R x accos( x )dx (1) R = KR with K = 1 pi ( α 3 cos3 (α) sin(α) 1 9 sin3 (α)) and α = accos( R ). Theefoe, fo any two points located in a cicle of adius R, the mean distance is: M(R) = 4KR (13) 5 Fom Eq. 9, Eq. 11 and Eq. 1, we can finally deduce the total elay taffic tot : tot = N paths P (x >) ( M 1) (14) The implementation of the methods peviously descibed has been ealized in Matlab.4. oth methods necessay etun the same esults that is efeed as theoy in Fig. 4. We can obseve that both the poposed mathematical fomulation and the theoetical fomulation match closely. Fig. 4. Numbe of paths.4 x implementation theoy density ssessment of the analytical method V. THROUGHPUT ESTIMTION: THE MULTIPLE SOURE-ESTINTION SENRIO In the pevious section, we demonstated that unde cetain conditions, namely when the paths do not intefee, the netwok pefomance in tems of thoughput and end-to-end delay can be significantly impoved. Howeve, this is without consideing the effects of concuent tansmissions between diffeent souce-destination pais. Let us look at the following scenaio with two flows fom to and fom to (Fig. 5). The nodes ae m apat, and the same taffic chaacteistics ae assumed fo both flows (R taffic at.5 packets/s and UP tanspot potocol) (a) Single Path Fig (b) Multipath Netwok topology with coss taffic To estimate the impact of the coss taffic on the nominal capacity, we detemine the maximum collision domains fo a single path and multipath scenaio. The bottleneck links ae epesented by lage solid aows in Fig. ). With a single path outing appoach, the maximal achievable thoughput is /8 wheeas in a multipath outing stategy, the uppe bound is as / (8 /+3) (a) Single Path Fig.. ollision omains (b) Multipath

6 We confimed these esults via simulations implemented in NS- (Fig. ). We can obseve that multipath outing still achieves slightly bette pefomance than single path outing. s expected, the thoughput fo both the single path scenaio and the multipath scenaio degades compaed to the pefomance esults obtained in Section III due to the inceased level of intefeence. The benefit of multipath outing ove single path outing becomes quite insignificant and is expected to vanish if the actual cost of the paths establishment is accounted fo. Thoughput (bit/s) Fig.. x 1 5 single path multipath Simulation time Impact of coss taffic on the thoughput VI. ONLUSION The need to povide pevasive communication at anytime, anywhee and the necessity to cope with the existing constaints of wieless envionments opened new eseach avenues to impove the netwok utilization and optimize the netwok capacity. Multipath outing has been put fowad as a potential solution to addess these issues, and many poposals have been popounded. Howeve, the poblem of intefeence has not been popely addessed, which questions the efficiency of these potocols in eal-wold deployments. In this pape, we focused on two elated poblems: 1) the estimation of the thoughput if only the intefeence of a single souce-destination pai is consideed; and ) the impact of intefeence when multiple souce-destination pais ae consideed. We povided an evaluation of the thoughput fo a -path outing scheme while accounting fo the intefeence of concuent data tansmissions fo a given souce-destination pai. We demonstated the complexity of the poblem and deived an analytical fomulation of the expected thoughput at each node consideing a unifom node distibution. We also poposed a simple method to establish multiple paths between a souce node and a destination node based on thei coodinates. With multiple souce-destinations pais, although a moe efficient netwok utilization due to a bette load balancing can justify the use of a multipath outing stategy compaed to single path outing, the benefits of multipath outing in tems of thoughput quickly vanish when intefeence is accounted fo. Theefoe, multipath outing can only be of inteest if the taffic distibution is known and if the effect of intefeence can be popely evaluated. Howeve, in most cases, assuming that such infomation is known o can be obtained easily is unealistic paticulaly due to the instability of netwok conditions in wieless envionments and the staleness of infomation (low convegence due to medium contention). In such context, multipath outing does not appea as a sound outing stategy. REFERENES [1].M. bbas and.n. Jain. Path diminution in disjoint multipath outing fo mobile ad hoc netwoks. In 15th IEEE Intenational Symposium on Pesonal, Indoo and Mobile Radio ommunications, volume 1, pages , Sep. 4. 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