WIRELESS Mesh Networks (WMNs) have recently attracted

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1 3968 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 8, AUGUST 23 A New MPLS-Based Forwarding Paradigm for Muti-Radio Wireess Mesh Networks Stefano Avaone and Giovanni Di Stasi Abstract Routing in muti-radio wireess mesh networks is a very chaenging probem. In this paper, we propose a forwarding paradigm based on MPLS (Muti Protoco Labe Switching) which makes use of a nove mechanism, denoted as MPLS spitting poicy. Such mechanism aows to configure mutipe next hops at an intermediate node, so that the incoming traffic is partitioned among the next hops according to predefined coefficients named spit ratios. The MPLS spitting poicy has been designed to aow for oad baancing and fast oca restoration. With such a mechanism, it is crucia to propery determine the set of spit ratios, as they determine how the traffic is routed across the network. We present an approach to compute a set of spit ratios that guarantee high performance under different traffic oads. To this end, we adopt the hose traffic mode, according to which we ony have knowedge of the imum amount of traffic entering or eaving the network at each edge node. A thorough simuation study is conducted to show that our approach outperforms other routing protocos in terms of throughput and robustness against traffic oad variations and singe node faiures. Index Terms Muti-radio wireess mesh networks, MPLS, routing. I. INTRODUCTION WIRELESS Mesh Networks (WMNs) have recenty attracted a big interest thanks to their abiity to wireessy cover arge areas with a ow cost of depoyment and maintenance. However, since wireess transmissions are invoved, interference is a major concern. In order to aeviate the interference, mesh routers are being equipped with mutipe radios which aow simutaneous transmissions on orthogona channes. The avaiabiity of mutipe radios per node eads to the channe assignment probem, i.e., the probem how to seect a channe for each radio in the network. Channe assignment is a chaenging probem and has been massivey investigated in the recent years [][2][3][4]. Routing in muti-radio wireess mesh networks is a very chaenging probem as we, due to a number of reasons: due to interference, nodes cannot dispose of the fu capacity of their inks, because the channe capacity must be shared among a the interfering inks. Thus, it does not suffice to ensure that each ink is not aocated more fow than its capacity to guarantee a feasibe routing. Instead, a routing protoco shoud be aware of which inks interfere with each other and route fows in such a way that the Manuscript received October 4, 22; revised March 4, 23; accepted June 8, 23. The associate editor coordinating the review of this paper and approving it for pubication was A. Abouzeid. S. Avaone and G. Di Stasi are with the Department of Computer Engineering, University of Napes, 825 Itay (e-mai: {stefano.avaone, giovanni.distasi}@unina.it). Digita Object Identifier.9/TWC /3$3. c 23 IEEE amount of fow aocated on each set of interfering inks does not exceed the channe capacity; wireess transmissions are affected by fading, propagation osses, environment noise, etc. Such phenomena may cause frequent (and temporary) unavaiabiity of inks. Therefore, a routing protoco shoud quicky react to such faiures by providing an aternate path to the destination; given the decrease in the avaiabe bandwidth caused by interference, it is necessary to fuy expoit the network resources. To this end, it is advisabe that a routing protoco support the abiity to route fows over mutipe paths between the same ingress-egress pair, in order to baance the traffic oad across the whoe network; the traffic oad offered to the network may vary dynamicay. A routing protoco shoud not be taiored for a particuar traffic matrix, but it shoud ensure a high performance despite variations in the traffic oad. To our knowedge, the routing protocos proposed so far for muti-radio wireess mesh networks fai to address a the issues mentioned above (see Section II). In this paper, we present a nove forwarding paradigm for muti-radio WMNs based on Muti-Protoco Labe Switching (MPLS) [5] with the purpose to address a the aforementioned issues. The first contribution of this paper is the definition of an MPLS spitting poicy, a new, standard-compiant, MPLS mechanism that enabes each intermediate node to spit the incoming traffic beonging to a specific Forwarding Equivaence Cass (FEC) among a predefined set of neighbors according to predefined spit ratios. As a resut, different packets of a given FEC foow distinct paths between the ingress and egress nodes. The proposed mechanism has the potentia for addressing the second and the third of the above issues. Indeed, the traffic between an ingress-egress pair can be baanced across mutipe paths (third issue). Aso, the avaiabiity of mutipe next hops for a given destination enabes a fast oca restoration in case of singe node/ink faiures (second issue). The proposed spitting poicy is presented in Section III. Enforcing the MPLS spitting poicy requires to: (i) identify a suitabe set of paths for each ingress-egress pair and (ii) compute the set of spit ratios. We address the former task by proposing RDAS (Resiient Directed Acycic Graph), an agorithm that finds, for a given ingress-egress pair, a set of paths that guarantee protection against singe node/ink faiures (Section V-C). To accompish the atter task, we present an approach to compute, given the current channe assignment, a set of spit ratios that ensure high performance despite variations in the traffic oad (Section V-B). To this end, rather than sticking to a given traffic matrix, we adopt the hose

2 AVALLONE and DI STASI: A NEW MPLS-BASED FORWARDING PARADIGM FOR MULTI-RADIO WIRELESS MESH NETWORKS 3969 traffic mode [6], according to which we ony have knowedge of the imum amount of traffic entering and eaving the network at each edge node, but we do not have knowedge of the actua traffic matrix. Based on such a mode, we formuate a convex optimization probem whose objective is to optimize the average performance over a the possibe traffic matrices (thus addressing the fourth of the above issues). The performance of a given soution is measured in terms of its abiity to route fows such that the amount of fow aocated on each set of interfering inks does not exceed the channe capacity (thus addressing the first of the above issues). The proposed convex optimization probem requires the knowedge of the network topoogy and of the imum amount of traffic entering and eaving the network at each edge node (hose mode). Given that mesh nodes are typicay stationary, changes in the network topoogy that require a re-computation of the spit ratios ony occur due to the addition/remova of mesh nodes (since temporary node/ink faiures are handed by the spitting mechanism). Aso, the imum amount of traffic entering (eaving) the network at an edge node can be set to a vaue cose to the sum of the transmission rates of the radio interfaces used for receiving (sending) the incoming (outgoing) traffic [6], hence such vaues are rather stabe as we. Thus, the computed set of spit ratios can be hed for a ong time. Hence, though the proposed approach is centraized, it entais a ow communication overhead because both the retrieva of topoogy information and the transmission of a new set of spit ratios to the network nodes do not need to be performed frequenty. The rest of the paper is structured as foows. In Section II we give an overview of the reated work. The MPLS spitting poicy is presented in Section III. In Section IV we formaize the probem to find a proper set of spit ratios for the use with the MPLS spitting poicy, whie in Section V we present our approach to sove such a probem. In Section VI we present the resuts of the simuation study we conducted to show that our approach achieves high throughput and is robust against variations in the traffic oad and against singe node faiures. Finay, Section VII concudes the paper. II. RELATED WORK Most of the work reated to routing in wireess mesh networks focused on ink or path metrics proposed as improvements upon the hop count metric. Among the first ink metrics to be introduced, the expected transmission count (ETX) [7] estimates the number of transmissions required to successfuy send a packet to a neighbor. The authors in [8] introduce two metrics, the expected transmission time (ETT) and the weighted cumuative ETT (WCETT). MIC (metric of interference and channe switching) [9] takes the interfow interference into account in addition to the intra-fow interference. A ink metric based on the estimated avaiabe bandwidth is instead proposed in []. The above mentioned ink metrics (and many others) are intended to be used with a singe path destination-based routing protoco. Very often, the routing protoco used to test the proposed routing metric is one of those designed for ad hoc networks, ike AODV [] or OLSR [2]. The routing protoco specified in the IEEE 82.s amendment [3], too, is basicay a modified version of AODV that uses the Airtime ink metric to associate each ink with an estimate of the amount of time needed to successfuy transmit a packet across that ink. These routing protocos, being singe path, have imited capabiities in terms of oad baancing and require some time to discover aternative routes in case of ink/node faiures. A number of proposas extend the above mentioned singe-path routing protocos to use mutipe paths between a source and a destination, such as AODV-BR [4], AOMDV [5] and AODV-DM [6].Such proposas define some measures of interference, but do not consider the avaiabe bandwidth resuting from the channe assignment. Aso, in case of faiures, repairing a path requires the exchange of routing messages and hence some time is needed to have consistent routing tabes. An adaptive oad-aware routing scheme is proposed in [7]. The network is divided into mutipe custers and each custer head estimates the traffic oad in its custer. If the estimated oad gets higher, the custer head increases the routing metrics of the routes passing through the custer so that the traffic avoids overoaded custers. This scheme requires a continuous adaptation of the ink costs to the offered traffic oad, which might ead to instabiities, and does not account for ink/node faiures. A number of approaches expoits the broadcast nature of the wireess medium. ExOR [8] is an opportunistic approach where a node broadcast a packet and the nodes that received it correcty agree on which of them has to further forward the packet, based on the distance to the destination. However, the protoco used to reach such agreement introduces some overhead. ROMER [9] buids a forwarding mesh around the minimum cost path and each packet is aowed to trave aong one of the paths in the forwarding mesh based on the current conditions. GATOR [2] is another opportunistic approach which expoits the knowedge of the geographic coordinates of the nodes whie seecting the receiver in charge of retransmitting a packet. A drawback of the opportunistic approaches, however, is that they are ess effective in mutiradio WMNs because ony the neighbors istening on the channe used by the sender can receive the packet. The anypath routing paradigm [2] generaizes the opportunistic approach. Each node is pre-configured with a set of next-hops, each having a different priority. A packet is further forwarded ony by the next-hop with the highest priority that correcty received the packet. An anypath is composed by a the possibe paths that a packet can take from the source to the destination. In [2] the goa is to find the east cost anypath, whie in [22] additive constraints to be satisfied are considered. Whie sharing some concepts with anypath routing, our proposa is deepy different. Firsty, anypath routing requires a modified MAC to determine which nexthop has to forward the packet. Secondy, the degree of oad baancing achieved depends on the outcome of the packet transmissions: if a the transmissions were successfu, the anypath routing woud reduce to singe-path routing. Thirdy, ike other opportunistic approaches, anypath routing is ess effective in muti-radio networks, where neighbors are reachabe via different channes. Finay, we mention our previous approach known as Layer- 2.5 forwarding paradigm (L2.5) [4]. In L2.5, forwarding decisions are not taken by ooking up the routing tabe, but

3 397 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 8, AUGUST 23 Fig.. Exampe to iustrate the spitting mechanism. MPLS SPLITTING({ρ i}, {ρ i },B,p) for each NHLFE i in decreasing order of ρ i ρ i 2 if the next hop in NHLFE i is reachabe 3 seect NHLFE i 4 ρ i = ρ i B + p B + p 5 ρ j = ρ j B j i B + p 6 B = B + p 7 if B>B 8 B = B min are based on two objectives: i) baance the traffic among the outgoing inks in proportion to their avaiabe bandwidth; ii) guarantee that a the packets reach the destination in a predetermined imum number of hops. L2.5 suffers from a oose contro over the paths taken by packets and fais to ensure that they are cyce-free. Aso, the performance of L2.5 is dependent on the avaiabe bandwidth vaues associated with the network inks, and hence it may degrade if they are not suited to the actua traffic oad. III. THE MPLS SPLITTING POLICY According to RFC 33 [5], MPLS nodes use three tabes to forward packets: NHLFE (Next Hop Labe Forwarding Entry), ILM (Incoming Labe Map) and FTN (FEC-to-NHLFE Map). An entry of the NHLFE specifies the next hop, the operation to perform on the packet s abe stack and (optionay) any additiona information needed in order to propery dispose of the packet. RFC 33 provides the foowing three operations: pop the abe at the top of the stack, push a new abe onto the stack, swap the abe at the top of the stack and possiby push other abes. The ILM is used when forwarding packets that arrive as abeed packets. An entry of the ILM maps an incoming abe to a set of NHLFE entries. The FTN, instead, is used when forwarding packets that arrive unabeed. An entry of the FTN maps a FEC (Forwarding Equivaence Cass) to a set of NHLFE entries. Normay, each FEC in a FTN entry is associated with a singe NHLFE entry and each abe in an ILM entry is associated with a singe NHLFE entry. In such a way, there is a unique next hop for a given FEC or abe at each node and thus a the packets of a FEC foow the same path (e.g., the dashed path from a to j in fig. ). As noted above, RFC 33 expicity mentions the possibiity that a abe or a FEC may be associated with a set of NHLFE entries, in order to perform, e.g., some sort of oad baancing. We expoit such a possibiity to aow the packets of a FEC to foow a predefined set of paths (as opposed to a singe path) between the ingress and egress nodes. Such an approach is iustrated by fig., where a continuous arrow departing from a node denotes a possibe next hop (as specified in an NHLFE entry) for the packets that entered the network at a and are destined to i. It can be observed that nodes have mutipe possibe next hops from among they seect the one which a given packet is forwarded to. Consequenty, different packets of the same FEC may foow distinct paths (e.g., a-d-h-i or a-b-d-f-i). A the possibe paths taken by the packets of a given FEC are determined a priori and can be enforced by propery configuring the MPLS tabes on the Fig. 2. Pseudo-code of the MPLS spitting poicy. nodes. For instance, the behavior of node b is achieved by configuring an entry in its ILM that associates the incoming abe 6 with two entries in the NHLFE: one that repaces abe 6 with abe 7 and sends the packet to node d and the other one that repaces abe 6 with abe 8 and sends the packet to node e. When an incoming packet with abe 6 arrives, node b has to seect either of the two NHLFE entries. In case a FEC or a abe is associated with mutipe NHLFE entries, the procedures to choose an NHLFE entry among the given set are beyond the scope of RFC 33. Here, we define a poicy to seect one of mutipe NHLFE entries that fits our goa to baance the traffic among the outgoing inks according to predefined proportions, whie ensuring a fast reaction to node/ink faiures. We assume that each NHLFE entry aso specifies, as an additiona information, a spit ratio, whichis a vaue between and. The spit ratios associated with a set of NHLFE entries that correspond to the same FEC or to the same abe must sum to. The goa of the spitting poicy is to baance the traffic matching a given FEC or a given abe among the neighbors specified by the corresponding NHLFE entries in proportion to the specified spit ratios. For this purpose, the agorithm shown in fig. 2 is used to seect an NHLFE entry (and hence a next hop) from among the set of NHLFE entries associated with a given FEC or with a given abe. The procedure shown in fig. 2 is given the set of the spit ratios {ρ i } associated with the set of NHLFE entries, the set of the actua utiizations {ρ i } of each NHLFE entry (i.e., the ratio of the amount of traffic transmitted as specified by an NHLFE entry to the tota traffic matching the FEC or the abe), a counter B that records the amount of traffic matching the given FEC or abe, and the size p of the packet for which an NHLFE entry must be seected. Before the traffic starts fowing, a the actua utiizations and the counter B are set to zero. Then, every time a packet matches a given FEC or abe, the associated NHLFE entries are sorted and visited in decreasing order of the gap between the spit ratio and the actua utiization. If the next hop neighbor incuded in the i- th NHLFE entry is marked as unreachabe, then the NHLFE entry is skipped. To this end, we assume that a feedback is provided by the ower ayers informing on the unavaiabiity of a neighbor. Otherwise, the packet is sent as specified by the i-th NHLFE entry and the actua utiization of a the NHLFE entries and the tota amount of traffic B are updated (ines 4 6). To avoid that B grows indefinitey, it is reset to a vaue B min once it exceeds a given threshod B. B min shoud be

4 AVALLONE and DI STASI: A NEW MPLS-BASED FORWARDING PARADIGM FOR MULTI-RADIO WIRELESS MESH NETWORKS 397 a vaue greater than zero to avoid that a the actua utiizations are reset after receiving the next packet (from ine 5, ρ j woud be nu if B were zero). Aso, a node that is marked as unreachabe can be incuded among the active neighbors again after a configurabe amount of time. Thus, the proposed MPLS spitting poicy enabes to baance the traffic matching a given FEC or abe among the neighbors specified in the associated NHLFE entries in proportion to the corresponding spit ratios. Aso, by having mutipe NHLFE entries (and hence next hop neighbors) aready configured, our spitting poicy enabes a fast restoration against singe node/ink faiures, as another NHLFE entry can be readiy used to send the packet. We note here that it aso makes sense to have NHLFE entries with an associated nu spit ratio. Such entries are not used to forward packets in norma conditions, but they are ony used in case a the other entries with non-nu spit ratios have been disabed due to the corresponding next hops being unreachabe. Thus, NHLFE entries with a nu spit ratio (that may be present in a soution returned by our approach proposed in Section V-B) can be usefuy configured since they serve as backup routes in case of faiures. IV. PROBLEM STATEMENT Interfering inks and coision domains: We assume that each mesh router is equipped with mutipe radio interfaces, each of which is assigned one of the C avaiabe channes and transmits at a fixed transmission power. We aso assume the avaiabiity of a set of transmission rates. Given that a radio may serve mutipe inks and the abiity of commodity hardware to set the transmission rate on a per-packet basis, we wi assign a rate to inks rather than radios. We mode the WMN as a directed graph G = (V,E), where V is a set of nodes each representing a mesh router. Given two nodes u, v V, a directed edge u v beongs to E iff u and v share at east a common channe and, in the absence of transmissions on other inks, there exists a rate r such that a transmission from u to v is successfu. We assume that a transmission from u to v is successfu if the Signa-to-Interference and Noise Ratio (SINR) at the receiver is sufficienty high to decode the signa. The SINR at receiver v when a signa is transmitted by u is defined as G SINR uv = uvp (u v) x y u v GxvP,whereP (u v) is (x y)+nv the power emitted by u to transmit to v, G uv is the gain of the radio channe between u and v, andn v is the therma noise at receiver v. Ifu transmits at rate r, the receiver v can correcty decode the signa if SINR uv γ r,whereγ r denotes the minimum SINR required to correcty decode a signa moduated at the rate r. Thus, a ink u v E G iff there exists a rate r such that uvp (u v) n v γ r. The highest rate r for which such inequaity hods is seected as the capacity of the ink and denoted by c(u v). In case u and v share mutipe channes, the set E may incude as many inks between the two nodes as the number of common channes. To differentiate among those inks and stress that a ink has been assigned channe c, we use the notation u c v. A ink x c y E interferes with u c v E if a transmission on x c y prevents a simutaneous transmission on u c v. We assume that happens when i) the two inks share the same transmitter or receiver, ii) the transmitter of a ink is the receiver of the other (since a singe radio cannot transmit and receive simutaneousy), or iii) a transmission on x c y makes the SINR at v too ow to correcty decode the signa from u. We define the set of a the inks that interfere with { u c v as its coision domain and denote it by D(u c v) = x c GuvP (u v) y E {x, y} {u, v} G xvp (x y)+n v <γ c c(u v) }. In other words, none of the inks in D(u c v) can be active at the same time as u c v. Finay, we define the tota utiization of the coision domain of ink e as U tot (e) = f(e ) e D(e) c(e, ) where f(e ) denotes the amount of fow routed on ink e. The MPLS spitting-based routing probem: Without oss of generaity, we consider a set V e = {n,...n N } V of N edge nodes acting as both ingress and egress nodes. We denote by {Is } s Ve and {Od } d Ve, respectivey, the sets of the imum amount of incoming and outgoing traffic at each edge node. According to the hose traffic mode, we ony have knowedge of these vaues and we know neither the actua amount of traffic entering at each edge node nor what portion of traffic entering at a given edge node is destined to each of the other N- edge nodes. A set of incoming fows I = {I s } s Ve is said to be feasibe if I s Is s V e. Our goa is to route the (unknown) traffic matrix, using MPLS and the spitting poicy, in such a way to minimize the cost function defined in Section V-A. A routing soution consists of a set of spit ratios {ρ s,d,where u v E s,d ρ s,d u v represents the ratio of the fow between the ingressegress pair (s, d) entering node u that is forwarded to node v and E s,d represents the set of inks aong which the fow between the ingress s and the egress d is routed. Ceary, the equation =must hod for each u and for ρ s,d u v v u v E s,d u v} s Ve,d Ve {s} each ingress-egress pair (s, d). The set of spit ratios determine how the (unknown) traffic matrix is routed across the network. Specificay, they determine, for each ingress-egress pair (s, d), a directed subgraph of G, S s,d =(V s,d, E s,d ),wherev s,d is the set of nodes beonging to the inks in E s,d.givenhow the spitting poicy works, it turns out that the packets fowing from ingress node s to egress node d can foow any of the paths between s and d in the subgraph S s,d. A routing soution is said to be admissibe if, for every ingress-egress pair (s, d), the set of inks E s,d, or, equivaenty, the directed subgraph S s,d, meets the foowing constraints: t ) t ) t ) to avoid that packets take excessivey ong paths, the ength of every path in S s,d must be at most α times the ength of the shortest path between s and d in G every path in S s,d must be cyce-free the set of paths in S s,d must guarantee protection against singe node/ink faiures, i.e., if a singe node/ink fais, the upstream node must have an aternative path to the egress node d We denote by fu v s,d the variabe representing the amount of fow between the ingress-egress pair (s, d) that is routed on ink u v. The tota amount of fow routed on a ink u v is f u v = d V e {s} fu v s,d. We denote by ϕs,d u v I s u v = f s,d the variabe representing the amount of fow on ink u v

5 3972 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 8, AUGUST 23 contributed by node s and destined to node d, normaized to the actua (unknown) amount of traffic I s entering source node s. We observe that ϕ s,d u v is independent of the actua amount of traffic entering source node s and ony depends on how traffic fows are routed. As shown in Section V-D, the set Φ={ϕ s,d u v} s Ve,d Ve {s} suffices to determine the set u v E s,d of spit ratios, and hence it can be considered as representative of a particuar routing soution. We denote by Γ(Φ, I) the cost of a particuar configuration where a routing soution Φ is used to route a feasibe set of incoming fows I = {I s } s Ve (see Section V-A for its definition). The average cost of a routing soution Φ, i.e., the average of Γ(Φ, I) over a the feasibe sets I, is denoted by: Γ(Φ) = Is n n N Γ(Φ, I)dI n di nn () Given the imum amount of traffic entering or eaving the network at each edge node, the MPLS spitting-based routing probem is to find a feasibe admissibe routing soution Φ that minimizes Γ(Φ). A routing soution is said to be feasibe, given {Is } s Ve and {Od } d Ve, if it obeys the foowing constraints: f ) the amount of fow routed on each ink must not exceed the ink capacity, for every feasibe set of incoming fows f ) the amount of fow routed towards each egress node must not exceed the imum amount of outgoing traffic of that egress node, for every feasibe set of incoming fows The feasibe admissibe routing soution that minimizes Γ(Φ) has the minimum cost on the average and hence it can be considered as the most robust routing soution against variations of the traffic matrix. V. SOLVING THE MPLS SPLITTING-BASED ROUTING PROBLEM In this section, we show how we sove the MPLS spittingbased routing probem defined in the previous section: we first define the cost Γ(Φ, I) of routing a given set I of actua incoming fows according to a particuar routing soution Φ. Then, we compute the average cost Γ(Φ) of a routing soution Φ over a the feasibe sets of incoming fows (Section V-A) then, we address the probem to find a feasibe admissibe routing soution minimizing Γ(Φ). Requiring that the returned routing soution be admissibe makes the probem to find a feasibe routing soution minimizing Γ(Φ) hard to sove. Hence, our approach is to decoupe the probem to find a set of directed subgraphs that make a routing soution admissibe from the probem to find a feasibe routing soution minimizing Γ(Φ) subject to the constraint that the fow between each pair of ingress and egress nodes can ony be routed aong the inks of predefined subgraphs. We present a convex optimization probem to find an optima soution to the atter probem (Section V-B) and propose a heuristic to sove the former probem (Section V-C) finay, we show how the set of spit ratios can be derived from the vaues of the ϕ s,d variabes (Section V-D) The main notations and symbos used in this section are summarized in Tabe I. A. Computing the average cost of a routing soution In this section, we define the cost Γ(Φ, I) of a particuar configuration where a given routing soution Φ = {ϕ s,d u v }s Ve,d Ve {s} is used to route a given feasibe set u v E s,d of incoming fows I = {I s } s Ve. The proposed cost function penaizes configurations where an excessive amount of fow is routed on the inks of a coision domain (which need to share the avaiabe channe capacity). To this end, we expoit the resuts of a theoretica anaysis [23] that shows that a set of fows routed on the inks of a coision domain satisfy the constraint on the channe capacity if the tota utiization of such coision domain is ower than a certain threshod λ (which depends on the overhead of the PHY ayer and has been determined in [23] for the case of IEEE 82.a). Aso, it has been shown through extensive simuations that: i) as ong as the tota utiization of a the coision domains is ess than λ, there is a high probabiity ( 85%) that a high percentage (>95%) of the traffic oad offered to the network is deivered to the destination; ii) if the tota utiization exceeds that threshod for some coision domains, the percentage of the traffic oad that is deivered to the destination is a decreasing function of the average tota utiization. Such resuts suggest that, in order to imize the portion of the traffic demands that is satisfied, we shoud strive to keep the tota utiization of a the coision domains beow a given threshod λ or, in case that is not feasibe, to minimize the average tota utiization. Hence, we define the cost of a particuar routing configuration as the average tota utiization over a the coision domains. However, in order to further penaize the soutions eading to high vaues for the tota utiization of some coision domains, we consider a weighted average of the tota utiizations. In particuar, we consider the weighting function: w(x) = ex e λ and define the cost Γ(Φ, I) of a particuar configuration as the average of w(u tot (e)) over a the inks e E: Γ(Φ, I) = E E w(u tot ( )) = E E e Utot() e λ (2) The weighting function is such that w(x) x if x λ and w(x) > x if x > λ, i.e., it decreases the weight of the tota utiizations beow λ and increases the weight of the tota utiizations above λ. The goa is thus to penaize the configurations with tota utiizations arger than λ. For conciseness, we define ϕ s = d V e {s} ϕs,d, i.e., ϕ s is the amount of fow on ink originated at node s (independenty of the destination node) and normaized to the actua (unknown) amount of traffic I s entering node s. It foows that the actua amount of fow routed on ink can

6 AVALLONE and DI STASI: A NEW MPLS-BASED FORWARDING PARADIGM FOR MULTI-RADIO WIRELESS MESH NETWORKS 3973 be expressed as Γ(Φ, I) = = ϕ s I s. Hence: E (e λ ) E (e λ ) e E e E D( ) s Ve s Ve D( ) ϕ s I s c() ϕ s I s c() Now, our goa is to compute Γ(Φ) by integrating Γ(Φ, I) over the region of a the feasibe sets of incoming fows (eq. ): Γ(Φ) = E = E n E (e λ ) n N E (e λ ) n n N e Is N Is N e s= s= D( ) D( ) ϕ s I s c() di di N ϕ s I s c() din di nn N s= I s (3) The integrating function is the product of N functions each depending on a distinct integration variabe. Hence, the mutipe integra can be decomposed as the product of N integras: n n N = = e D( ) D( ) D( e ) ϕ s c() e ϕ s I s c() din di nn D( ) ϕ s I s c() D( ) ϕ s c() ϕ s I s c() I s Hence, the average cost Γ(Φ) of a routing soution Φ over a the feasibe sets of incoming fows is: ϕ s I s Γ(Φ)= e λ E D( e ) c() Is ϕ s s V E e D( ) c() (4) B. Finding an optima feasibe routing soution The average cost Γ(Φ) of a particuar routing soution Φ over a the feasibe sets of incoming fows is expressed by equation (4). Here, we formuate a convex optimization probem, denoted as METER (Minimum average cost feasibe Routing), to find a routing soution that is feasibe given the Notation D() c() V e Is Od I s I E s,d f s,d f ϕ s,d ϕ s Φ Γ(Φ, I) Γ(Φ) ρ s,d u v TABLE I NOTATIONS AND SYMBOLS USED IN THIS PAPER Expanation coision domain of ink E capacity of ink E subset of nodes acting as ingress/egress nodes imum amount of incoming traffic at s V e imum amount of outgoing traffic at d V e actua (unknown) amount of incoming traffic at s V e {I s} s Ve set of inks aong which fow between s and d is routed fow between edge nodes s and d that is routed on ink tota fow routed on ink fow between s and d on ink (f s,d ) normaized to I s fow originated at s and routed on ink, normaized to I s routing soution (set of ϕ s,d for each s, d, E s,d ) cost of routing a given set of incoming fows according to Φ average of Γ(Φ, I) over a feasibe sets of incoming fows I ratio of the fow from s to d entering u that is forwarded to v imum amount of traffic entering or eaving the network at each edge node and minimizes Γ(Φ), subject to the constraint that the fow between each pair of ingress and egress nodes can ony be routed aong a predefined set of inks. Soving such a probem provides the normaized amount of fow ϕ s,d routed on each ink and beonging to each ingress-egress pair (s, d). From such information, as iustrated in section V-D, we can derive the set of spit ratios that each node needs to enforce our MPLS spitting poicy. The formuation of the METER probem is shown in fig. 6. Besides the set of normaized variabes {ϕ s,d } s Ve,d Ve {s}, we aso consider a set of auxiiary E s,d variabes {F s,d } s Ve,d Ve {s}, each representing the amount of fow routed between an ingress node s and an egress node d, normaized to the actua incoming traffic at node s. The objective of METER is to minimize Γ(Φ). Constraints ) represent the usua (normaized) fow conservation constraint that must be enforced at each node for every pair of ingressegress nodes. Constraints 2) ensure that a the actua amount of incoming fow at each edge node is spit among the other edge nodes. Constraints 3) ensure that the amount of fow routed towards each egress node does not exceed the imum amount of outgoing traffic of that egress node, for every feasibe set of incoming fows (constraint f of Section IV). Indeed, if the incoming set of fows is feasibe, then F s,d I s F s,d Is,wheretheeft {d} {d} hand side is the actua amount of fow routed towards egress node d. Hence, if constraint 3) hods, constraint f ) hods as we. Constraints 4) prevent the incoming (outgoing) fow at an edge node to be re-routed back to the ingress node (from the egress node). Constraints 5) ensure that the amount of fow routed on each ink does not exceed the ink capacity, for every feasibe set of incoming fows (constraint f of Section IV). Finay, constraints 6) ensure that the fow between edge nodes s and d is ony aocated on inks that beong to the predefined set E s,d. In such a way, if the predefined set of inks are propery computed, the routing soution returned by the optimization probem is guaranteed to be admissibe. In Section V-C we present an agorithm that finds, for a given ingress-egress pair (s, d), a directed subgraph that meets

7 3974 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 8, AUGUST 23 Fig. 3. Exampe to iustrate oops in the directed subgraph Fig. 4. Exampe to iustrate paths exceeding the imum ength in the directed subgraph Fig. 5. Topoogicay sorted nodes of the DAG in fig. 4. variabes ϕ s,d [, ] E, s V e, d V e {s} F s,d [, ] s V e, d V e {s} ( { } ) s Ve,d Ve {s} minimize Γ ϕ s,d E s,d subject to ) ϕ s,d u v ϕ s,d u v E s,d v u E s,d if u s u d F s,d if u = s F s,d if u = d 2) 3) d V e {s} {d} 4) ϕ s,d u v = 5) Fig. 6. F s,d = F s,d I s d V e {s} 6) ϕ s,d = O d v u = u V, s V e, d V e {s} s V e d V e s V e, d V e {s}, v= s u = d ϕ s,d Is c() E Formuation of the METER probem. s V e, d V e {s}, / E s,d constraints t ), t )andt ) of Section IV. METER is a convex optimization probem, because the objective function is convex (see the Appendix) and the constraint functions are inear. Convex optimization probems have the property that a ocay optima point is aso gobay optima. Hence, we use an interior point method [24] to find an optima soution to the probem. C. Finding a directed subgraph for an ingress-egress pair We now address the probem to find, for a given ingressegress pair (s, d), a set of inks E s,d or, equivaenty, a directed subgraph S s,d that guarantees that a routing soution is admissibe. The approach we foow is to find a set of paths between s and d (in G) and insert a of their inks into E s,d. Given the constraint on the imum aowed ength of any path between s and d in S s,d, a possibe approach woud be to use a k-shortest oopess path agorithm [25] to find a the oopess paths between s and d in G having a ength ess than the imum one. A k-shortest oopess path agorithm indeed returns a the shortest paths in increasing order of ength. However, the foowing two issues shoud be considered: ony adding oopess paths to E s,d does not ensure that a the possibe paths in the subgraph S s,d are oopess. An exampe is iustrated in fig. 3, where the inks of two oopess paths (a-c-g-h-f-i and a-b-e-f-d-h-i) are added to E s,d. The resuting subgraph incudes paths (e.g., a-b-e-f-d-h-f-i) containing a cyce (f-d-h-f) ony adding paths with ength ess than the imum aowed one does not ensure that a the possibe paths in the subgraph have a ength ess than the imum aowed one. An exampe is iustrated in fig. 4, where the imum ength is fixed to 6 hops. If we add the inks of two paths satisfying the constraint on the imum path ength (a-d-c-g-j-i and a-b-e-f-d-h-i), the resuting subgraph incudes a path (a-b-e-f-d-c-g-j-i) havinga ength (8 hops) exceeding the imum aowed ength Therefore, we present an agorithm that finds, for a given ingress-egress node pair, a directed subgraph of the network topoogy that meets constraint t ), t )andt ) of Section IV. Basicay, the subgraph is initiaized to contain the shortest path between the ingress and egress nodes and then it is augmented with other paths to fufi constraint t ). Every time we attempt to add a path to the subgraph, we check whether the augmented subgraph contains a cyce or a path having a ength exceeding the imum one. Fortunatey, the subgraph we seek is a directed acycic graph (DAG) and hence performing the above checks is as simpe as running a Depth- First-Search (DFS) [26]. Aso, a DFS in a DAG aows to sort the nodes in a topoogica ordering, which is such that if a ink u v exists in the DAG, then u precedes v in the ordering. Thus, the ingress (egress) node is the first (ast) node in this ordering. As an exampe, fig. 5 shows a topoogica sort ordering of the DAG resuting from the two paths highighted in fig. 4. Figure 7 shows the pseudo-code of the Depth-First-Search visit used by our agorithm. The DFS procedure initiaizes the attributes (coor, predecessor and imum distance from the egress node) of a the nodes of the subgraph D, cears the ist that wi contain the nodes sorted in a topoogica ordering, sets the booean variabe acycic to true and cas DFS VISIT on node s. TheDFS VISIT procedure performs the cassic DFS visit starting from node u. In addition, if a back edge is detected (ines 6 7), the acycic variabe is set to fase. Indeed, a directed graph is acycic if and ony if a depthfirst-search yieds no back edges. To obtain a ist of the nodes of the DAG sorted in a topoogica ordering, we can push

8 AVALLONE and DI STASI: A NEW MPLS-BASED FORWARDING PARADIGM FOR MULTI-RADIO WIRELESS MESH NETWORKS 3975 DFS(D =(V D,E D),s) for each u V D 2 coor [u] =WHITE 3 pred[u] =NIL 4 dist[u] = 5 sortedlist = 6 acycic = TRUE 7 DFS VISIT(D, s) DFS VISIT(D =(V D,E D),u) coor [u] =GRAY 2 for each v V D u v E D 3 if coor [v] =WHITE 4 pred[v] =u 5 DFS VISIT(D, v) 6 ese if coor [v] =GRAY 7 acycic = FALSE 8 coor [u] =BLACK 9 if sortedlist dist[u] = v u v E D dist[v]+ sortedlist = u, sortedlist Fig. 7. Pseudo-code of the depth-first-search. nodes on the front of such a ist as soon as they are marked as back (ine ). Aso (ines 9 ), when inserting a node u in such a ist (but the first node being inserted, which is the egress node), we compute the imum distance in the DAG between u and the ast node of the sorted ist (the egress node) by increasing by the imum distance of each neighbor of u (at this point, a the neighbors v of u such that u v exists in the DAG have been aready inserted into the sorted ist, by definition of topoogica ordering). Therefore, a DFS on a subgraph D checks whether D is a DAG and, in that case, returns a sorted ist of nodes in a topoogica ordering and the ength of the ongest path between each node in D and the ast node in the topoogica ordering (the egress node). We now present our agorithm, denoted as RDAS (Resiient Directed Acycic Subgraph), to find, for a given pair (s,d) of ingress and egress nodes, a directed subgraph of a graph G that meets constraint t ), t )andt ) of Section IV. Basicay, RDAS (fig. 8) initiaizes the subgraph D to the shortest path in G between the ingress and egress nodes and then expores the nodes in D in a reverse topoogica order, starting from the penutimate node. To satisfy constraint t ), an attempt is made to ensure that the expored node u has two distinct next hops in D. To this end, a path between u and the egress node d is sought that does not incude the current next hop of u and satisfies t )andt ). If such a path is found, it is added to D and the exporation restarts from the penutimate node in the new topoogica ordering of the nodes in D. An expored node is marked as done, so that it is expored just once. The agorithm ends when the ingress node is marked as done. We now describe the exporation of a node in more detais. If the expored node has been aready marked as done, we continue by exporing its predecessor in the topoogica ordering of the nodes in D (ines 9 ). If the expored node has aready more than one next hop in D, it is marked as done and its predecessor in the topoogica ordering is then expored (ines 34 35). If the expored node u has a singe neighbor RDAS(G =(V,E),s,d,α) D = 2 SP = SHORTEST PATH(G, s, d) 3 PATH ADD(D, SP) 4 DFS(D, s) 5 for each u V 6 done[u] =FALSE 7 u = previous[back [sortedist[d]]] 8 whie u<>nil 9 if done[u] =TRUE u = previous[u] continue 2 if Adj (D, u) = 3 v = front[adj (D, u)] 4 G Pruned = G 5 if v = d 6 REMOVE EDGE(G Pruned,u d) 7 ese REMOVE VERTEX(G Pruned,v) 8 found = FALSE 9 L su = MAX DIST FROM SOURCE(D, u) 2 L ud = 2 whie!found AND L su + L ud α ength[sp] AND KSP HAS NEXT(G Pruned,u,d) 22 P = KSP NEXT(G Pruned,u,d) 23 D Augm = D 24 PATH ADD(D Augm,P) 25 DFS(D Augm,s) 26 if IS ACYCLIC(D Augm) AND MAX DIST TO DEST(D Augm,s) α ength[sp] 27 found = TRUE 28 L ud = ength[p ] 29 if found 3 done[u] =TRUE 3 D = D Augm 32 u = previous[back [sortedist[d]]] 33 continue 34 done[u] =TRUE 35 u = previous[u] Fig. 8. Pseudo-code of the RDAS agorithm. (v) ind, we attempt to find an aternative path to the egress node. For this purpose, we consider a copy (G pruned )ofthe input graph G and prune the ink u v, in case v is the egress node, or the vertex v otherwise. Then, we ook for a path between u and the egress node d in the pruned graph. A k-shortest oopess path agorithm [25] provides, one-by-one and in increasing order of ength, the shortest paths between u and d in the pruned graph. The path P returned by the k- shortest path agorithm is tentativey added to a copy (D Augm ) of the subgraph D. ADFSofD Augm is run, which determines whether D Augm is acycic, finds the ength of the ongest path between s and d and topoogicay sorts the nodes. If D Augm is acycic and the ength of the ongest path is ess than the imum aowed path ength, the path P is actuay added to D, node u is marked as done and the exporation of the nodes restarts from the penutimate node in the new topoogica ordering (ines 29 33). Otherwise, a new path returned by the k-shortest path agorithm is considered. To avoid that a number of shortest paths between u and d in the pruned graph are useessy considered, we compute the ength L su of the ongest path between the ingress node s and u in D (ine 9). Given that we aready have a topoogica

9 3976 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 8, AUGUST 23 (a) Sampe vaues for the ϕ s,d variabes (b) Corresponding set of spit ratios Fig. 9. Deriving the set of spit ratios from the set of ϕ s,d variabes. ordering of the nodes in D, computing such a vaue ony requires to reax a the edges of D (whose weights must be set to -) [26]. Thus, as soon as the k-shortest path agorithm returns a path with a ength L ud such that L su + L ud exceeds the imum aowed path ength, we can stop processing the shortest paths between u and d. Indeed, since shortest paths are returned in increasing order of ength, none of the foowing shortest paths can be added to the subgraph without vioating the constraint on the imum path ength. In such a case, node u is marked as done (despite it ony has one neighbor) and the predecessor of u in the topoogica ordering is expored. The agorithm ends when the ingress node s is marked as done and returns the directed subgraph D. The compexity of RDAS is dominated by the inner whie oop (ines 2 28). In the worst case (since the nodes in D are a subset of those in G), a DFS on D requires O( V + E ), whie obtaining the next path from the k-shortest path agorithm requires O( V ( E + V og V )). The outer whie oop is repeated at most V times (once for each node in D), hence the compexity of RDAS is O( V 2 ( E + V og V )). D. Computing the optima set of spit ratios Soving the METER probem (fig. 6) provides the vaues forthesetofvariabes{ϕ s,d } s Ve,d Ve {s} representing the E s,d amount of fow routed on each ink and associated with each pair of ingress-egress nodes, normaized to the actua incoming fow at the ingress node. Figure 9a shows a network with some sampe vaues for the fows routed between a and i and between a and j (the superscript next to a vaue indicates the destination of the fow) normaized to the incoming fow at a. Since the spitting poicy baances the traffic among the outgoing inks in proportion to the spit ratios, to achieve the normaized fows given by the set of variabes we need to set the spit ratios as foows: {ϕ s,d } s Ve,d Ve {s} E s,d ρ s,d u v = ϕ s,d u v ϕ s,d u w u w E s,d Figure 9b shows the set of spit ratios corresponding to the normaized fow vaues of fig. 9a. VI. PERFORMANCE EVALUATION We conducted a number of simuation studies to evauate the performance of the proposed MPLS-based forwarding paradigm. For this purpose, we impemented a software too that uses the panar coordinates of the mesh nodes to buid the Fig.. Empirica CDF of δ. network topoogy based on the interference mode described in Section IV. We assume the gain G uv of the radio channe between u and v to be the reciproca of the square of the distance between u and v and the therma noise to be -2dbm. The SINR threshods are set to aow a rate of 54Mbps when the nodes are within 3m, 48Mbps when within 32m, 36Mbps when within 37m, 24Mbps when within 45m, 8Mbps when within 6m, 2Mbps when within 69m, 9Mbps when within 77m and 6Mbps when within 9m. The channe assignment is performed by FCRA [4]. Since FCRA is a traffic-aware channe assignment agorithm, different channe assignments can be obtained by feeding FCRA with different sets of ink fows. We consider two topoogies with 3 nodes and two radios per node. In the dense topoogy, nodes are paced in a 93 25m 2 area and the imum ink rate is 54Mbps. In the sparse topoogy, nodes are paced in a m 2 area and the imum ink rate is 36Mbps. We consider 4 edge nodes and hence 2 ingress-egress pairs. METER, the convex probem described in Section V-B, is soved by using the open source software Ipopt (Interior Point OPTimizer). The next subsections (but the first one) aim to evauate the performance of our approach in terms of network throughput. To this end, experiments were carried out with the network simuator ns-3. We contributed to the impementation of the MPLS modue in ns-3 and added the MPLS spitting poicy In such experiments, we compare our approach (simpy abeed as MPLS) based on the MPLS spitting poicy with the spit ratios determined as shown in Section V to our previous Layer- 2.5 forwarding paradigm (L2.5) and to the routing protoco specified in IEEE 82.s [3]. Uness otherwise stated, RDAS is run with α=3 in order to consider paths that are much onger than the shortest path, which ikey consists of inks between distant nodes utiizing ow bit rates. The threshod λ is set to.5. In the ns-3 experiments, TCP traffic is generated (for a duration of 6 seconds) according to the on-off mode, with T on U(.5s,.5s) and T off U(.5s,.5s). A. Comparing METER-RDAS to a ower bound Our approach to sove the MPLS spitting-based routing probem is to sove the METER-RDAS probem, where RDAS is used to compute the set of directed subgraphs that are As of this writing, the MPLS modue has not been merged yet into the mainine ns-3 code. The MPLS code is avaiabe at

10 AVALLONE and DI STASI: A NEW MPLS-BASED FORWARDING PARADIGM FOR MULTI-RADIO WIRELESS MESH NETWORKS 3977 Fig.. (a) Dense topoogy, 3 channes (b) Dense topoogy, channes (c) Sparse topoogy, channes Average throughput achieved under different traffic oads. required to formuate the METER probem. Thus, our approach might not return the optima feasibe admissibe routing soution of the MPLS spitting-based routing probem, because the returned routing soution is constrained to aocate fow on the inks of the directed subgraphs computed by RDAS. Though the average cost of the optima routing soution Φ opt is difficut to find, it is straightforward to compute a ower bound to such vaue. To this end, we denote by E s,d KSP the set of inks of a the paths between s and d in G whose ength is at most α times the ength of the shortest path. Such a set can be easiy computed by using a k-shortest oopess path agorithm to find a the paths between s and d with ength ess than the imum one. E s,d KSP is not guaranteed to satisfy constraints t ), t )andt ), but it certainy incudes any set E s,d eading to an admissibe routing soution. Thus, if we sove the METER probem with E s,d KSP as the set of inks that are aowed to carry the fow between s and d (we denote such a probem by METER-KSP), the obtained objective vaue, denoted as Γ(Φ KSP ), represents a ower bound to the average cost of the optima feasibe admissibe routing soution. In order to compare the average cost of the routing soution returned by METER-RDAS, denoted as Γ(Φ RDAS ),tothe ower bound to the minimum average cost, we performed experiments with varying vaues for the imum amount of traffic entering and eaving the network at the edge nodes, different channe assignments and α vaues (ranging from.5 to 3). For each experiment, we soved both METER-RDAS and METER-KSP and computed δ = Γ(ΦRDAS ) Γ(Φ KSP ) Γ(Φ KSP ), i.e., the percentage increase with respect to the ower bound. The empirica CDF (Cumuative Distribution Function) of the vaues of δ resuting from our experiments is shown in fig.. It can be observed that the percentage increase of the average cost of the routing soution returned when using RDAS is aways beow 2%, whie the percentage increase is beow % in the 7% of the cases and beow 6% in the 3% of the cases. If we consider that such resuts refer to a comparison with a ower bound and that E s,d KSP is unikey to ead to an admissibe routing soution, we can assert that our approach achives an average cost very cose to the minimum one. B. Robustness against variations in the traffic oad Our approach to sove the MPLS spitting-based routing probem has been designed to provide a set of spit ratios ensuring high performance under different traffic oads. The experiments described in this section aim to show that our approach is actuay more robust against variations in the traffic oad than other routing protocos such as the defaut routing protoco specified in IEEE 82.s and our previous approach L2.5. We report the resuts obtained for the two topoogies mentioned earier, where the imum amount of traffic entering each edge node is uniformy distributed between 6Mbps and 8Mbps. We aso considered the avaiabiity of 3 and orthogona channes. For each topoogy, we performed different experiments where the actua traffic oad entering the network was, on the average, a percentage of the imum amount ranging from 2% to 6%. For each such cases, we considered 4 uniform random variabes with a mean equa to the required percentage (e.g., U(.4,.6), U(.3,.7), U(.2,.8), U(.,.9) in case the actua traffic oad is, on the average, the 5% of the imum amount). Each of such uniform random variabes was used to derive the fraction of the imum amount of traffic entering each edge node that actuay entered the network. Then, the actua traffic entering each edge node was spit among the destination nodes in 5 different ways, thus eading to a tota of 2 experiments for each topoogy and for each given percentage of the imum amount of traffic oad. Figure summarizes the distribution of the throughput (average over the whoe duration of a simuation) achieved by each agorithm in the 2 experiments carried out for each given percentage of the imum amount of traffic oad. In particuar, a vertica ine spans from the minimum to the imum vaues, whie a white box spans from the first quartie to the third quartie. It can be observed that our approach outperforms 82.s and L2.5 in a the considered scenarios. The average throughput achieved by our approach is indeed from 2% to 2% higher than that of the other routing protocos. The poor performance of 82.s in some experiments can be expained by considering that 82.s is a singe path routing protoco and therefore, in case of a high traffic demand between an ingress-egress pair, the seected path may be easiy congested, thus eading to a decrease in the throughput. The poor performance of L2.5 in some experiments can be expained by considering that each node attempts to utiize each ink in proportion to predefined fow rates, that are returned by a traffic aware channe assignment agorithm. Thus, if the channe assignment has been computed based on a traffic oad which is different than the actua offered

11 3978 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 8, AUGUST 23 Fig. 2. (a) Dense topoogy, 3 channes (b) Dense topoogy, channes (c) Sparse topoogy, channes Transferred data oss due to a singe node faiure. oad, the performance of L2.5 may decrease. Figures a-c aso show that our approach ensures higher robustness against variations in the traffic oad. Indeed, the ratio of the imum average throughput to the minimum average throughput achieved in the 2 experiments associated with a given topoogy and a given percentage of the imum amount of traffic oad ranges from 5% to 2% for our approach, from 23% to 4% for 82.s and from 5% to 6% for L2.5. Such resuts prove that the performance in terms of throughput of our approach keeps firmy at high vaues despite variations in the traffic oad, whie the performance of the other routing protocos osciates between ow and moderate to high vaues depending on the offered traffic oad. C. Robustness against a singe node faiure The experiments described in this section aim at evauating the behavior of the routing protocos in the presence of a singe node faiure. We consider the same topoogies as in the previous section and an actua traffic oad equa to 2% of the imum traffic oad. Each experiment asts 45 seconds and, after 5 seconds from the beginning, a node faiure is simuated by increasing the noise eve at every radio interface of that node to the point that the node is not abe to send or receive packets. For each topoogy, we perform 26 experiments, each invoving the faiure of a different node (except the edge nodes), and compute, for each s interva, the average throughput over a the experiments. The average throughput over a s interva represents the amount of data transferred to the destination nodes during that interva. Figures 2a-2c show, for each second foowing the faiure, the ratio of the average (over the 26 experiments) amount of data transferred to the destination nodes since the faiure to the amount of data transferred in the same interva in the absence of faiures. It can be observed that our approach based on the MPLS spitting poicy outperforms both 82.s and L2.5. Indeed, our approach keeps the data oss in case of a node faiure beow 5% in a the considered scenarios. Instead, 82.s and L2.5 experience higher data osses, ranging from 5% to 8%. Such resuts thus prove that our approach is abe to better react to singe node faiures than 82.s and L2.5. VII. CONCLUSION We addressed the probem to deveop a routing strategy for muti-radio wireess mesh networks which: i) takes the constraint on the channe capacity into account; ii) isabeto quicky recover from node/ink faiures; iii) supports mutipath routing; iv) ensures high performance with a wide range of traffic matrices. For this purpose, we presented a nove mechanism, the MPLS spitting poicy, which consists in aowing mutipe candidate next hops at each intermediate node for a given FEC and partitioning the traffic of that FEC among such next hops in proportion to predefined spit ratios. The MPLS spitting poicy enabes to baance the traffic oad across mutipe paths and aows for a fast oca restoration. We then deveoped a technique to compute the set of spit ratios in order to ensure high throughput despite variations in the traffic oad. This goa is achieved by propery defining a cost function, computing directed subgraphs aong which the fow of each ingress-egress pair can be routed and soving a convex optimization probem. Finay, we performed a thorough simuation study which confirmed that our approach outperforms other routing protocos in terms of network throughput and robustness against oad variations and singe node faiures. APPENDIX We first show that f : x R N N i= ex i x i is ogconvex, i.e., og f( x) = N i= og ex i x i is convex. To this end, we denote by g i ( x) = og ex i x i the i-th term in the expression of og f( x). Since 2 x j x k g i ( x) =if j i or k i, it foows that the Hessian of g i ( x) is a diagona matrix: ( 2 g i ( x) =diag x 2 i e xi,,... (e xi ) 2 The unique non-nu eigenvaue is aways positive (it can be easiy seen by potting its graph) and hence the Hessian is positive semidefinite, which means that g i ( x) is a convex function. Being a sum of convex functions, og f( x) is convex, i.e., f( x) is og-convex, which impies that f( x) is convex. We now consider the coumn vector ϕ R N D() : ϕ =[ϕ...ϕ d ϕ 2...ϕ 2 d...ϕ N...ϕ N d ] T where is a ink, d = D( ) and { i } d i= = D( ), andthe matrix A R N N D() : a,... a,d a 2,d+... a 2,2d A = a N,(N )d+... a N,Nd )

12 AVALLONE and DI STASI: A NEW MPLS-BASED FORWARDING PARADIGM FOR MULTI-RADIO WIRELESS MESH NETWORKS 3979 where Ii if (i )d <j id, a i,j = c( (j ) mod d+ ) otherwise with i =,...d and j =,...Nd. Then, the function: h( ϕ )=f(a ϕ )= N e s= D( ) I s ϕ s I s c() ϕ s c() D( ) is convex because obtained from a convex function (f( x)) through composition with an affine mapping. Hence the average cost of a routing soution (eq. 4) can be expressed as: Γ(Φ) = e λ [ E ] h( ϕ ) E and therefore it is convex as we, because sum of convex functions. Hence, the objective function of the optimization probem defined in section V-B is convex. ACKNOWLEDGMENT This work was partiay funded by the Itaian Ministry of Education, University and Research (MIUR) within the framework of project PRIN 29 Software router to Improve Next-Generation Internet (SFINGI). REFERENCES [] A. Raniwaa, K. Gopaan, and T. 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Yen, Finding the K shortest oopess paths in a network, Management Science, vo. 7, no., pp , Juy 97. [26] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Agorithms, 2nd ed. MIT Press, 2. Hoc Networks. Stefano Avaone received the M.S. degree in Teecommunications Engineering (2) and the PhD degree in Computer Networks (25) from the University of Napoi Federico II. He is currenty an Assistant Professor with the Department of Computer Engineering at the University of Napoi. His research interests incude wireess mesh networks, traffic engineering, QoS routing. He was a visiting researcher at the Deft University of Technoogy (23-4) and at the Georgia Institute of Technoogy (25). He is on the editoria board of Esevier Ad Giovanni Di Stasi received the Laurea degree in Computer Engineering (27) and the PhD degree in Computer Networks (2) from the University of Napoi Federico II. He was a visiting researcher at INRIA Sophia Antipois, France (29) and at the Karstad University, Sweden (2). His current research interests incude experimenta reserch infrastructures and testbeds, routing and channe assignment agorithms for wireess mesh networks and peer-to-peer traffic optimization.