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Capacity Assignment for Multipath Routing in Multiclass Two-tier Wireless Mesh Networks
Abdelhak Farsi, Nadjib Achir, Khaled Boussetta
L2TI – Institut Galil´ ee – University Paris 13 99 Avenue J-B Cl´ ement, 93430 Villetaneuse, France Emails: {farsi, nadjib.achir, khaled.boussetta}@univ-paris13.fr the choice of single path to achieve the destination node [5]. The option to have more than one radio interface per mesh node became later a mandatory, the reason that urged other works to deal with Multi-radio and multi-channel wireless mesh networking [6]. Among the best alternatives to optimally maximize the utilization of network resources, is to consider several routes to achieve a destination. Multipath routing can provide the best performances over single flow routing, when mesh nodes within coverage area of each other avoid interference and communicate over orthogonal channels using sufficient radio interfaces. In practice, selection of multiple routes depends on the assigned channel, if the used channels are not orthogonal, the mesh node interferes with it self. Authors in [7] formulate the multipath routing problem throughout a linear programming system to optimize the network throughput by resolving jointly channel assignment and link scheduling. In [8] the authors propose a channel aware routing metric for multipath selection in multi-radio networks. The advantage of multipath routing has been argued in [9], where simulation results have clearly shown its reliability over single path routing. Our contribution for the aforementioned WMN Dimensioning can be summarized as follow: We first model mobility at the access tier and classify users based on their running applications into two kinds of classes according to the constant or the minimal bandwidths required. Depending on this classification that it will be used as input for two different strategies, detailed bellow, in order to guarantee exactly the needed bandwidth for delay-sensitive applications and ensure a minimal bandwidth for the non delay-sensitive applications. Another contribution is to maximize the load of the WMN by taking advantage of multicommodity concept to derive the fairness multipath routing protocol. The rest of this paper is organized as follows: Section II discusses the considered network topology, the assumptions that we based on and our contributions. Section III details the used strategies to share the capacity at the access tier. Section IV explains how we can optimize the concurrent flows at the backhaul tier following different routes where a modified Maximum Concurrent Flow model is presented herein. Section V shows the obtained numerical results. Finally the paper is concluded in section VI.

Abstract— Nowadays, Wireless Mesh Networks (WMN) are increasingly supporting jointly non real-time and real time applications that required a guaranteed bandwidth. A new methodology is presented throughout this study to show that an important aspect of dimensioning WMNs, considering mobility of users, fairness and their demand according to two types of running applications. This leads to satisfying voice demand while ensuring fairness among the non delay-sensitive applications. We propose the combination of both capacity estimation based on users mobility and multipath scheme. We compare our proposition with single path routing based on the shortest path criteria in terms of end-to-end throughput to show that it performs best.

Keywords: Wireless Mesh Networks, Capacity assignment, Complete Partitioning, Multi-Concurrent Flow, Fairness I. I NTRODUCTION Despite recent progresses in WMN, there are various works dealing with challenging aspects. We have focussed our study on dimensioning and routing themes to design a reliable and efficient WMN. Dimensioning has not been deeply studied in WMNs, but since it has commun objectives as deployment, it has been treated in aroundabout way. The optimal position of the gateway and mesh nodes in WMN [1] is crucial to balance the load, ensure the optimal path, and minimize the number of gatways. Authors in [2] deal with deployment in WMN by determining the cost of the topology, the tower height and the used antenna. The overall problem has been divided up into four tractable problems. The placement of Mesh nodes and gateways have been studied in [3] and [1] respectively. In the former reference, the purpose was to find the suitable location which ensures both coverage and connectivity. However, the latter reference considered only the gateway placement to maximize the overall throughput and ensures fairness among all mesh nodes on top of scheduling communications. Authors in [4] formulate dimensioning problem, related to the elastic traffic, for WMN taking into account both interference and the flow-level performance to estimate the required amount of resources. However, three limitations could be noted here, the traffic nature, the scheduling strategy in addition to the use of single channel single radio interface. Routing in wireless networks has been much investigated and numerous approaches have been designed and implemented. Almost all of them based on end-to-end metrics and
This work was supported by grants from R´ egion Ile-de-France

978-1-4244-5626-0/09/$26.00 ©2009 IEEE

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The considered two-tier WMN model

II. P ROBLEM STATEMENT A. Wireless Mesh Network model As illustrated in fig 1, we consider a two-tier WMN with stationary wireless routers, where each router is equipped with a certain number of radio interfaces. Among all nodes, only one, noted rg , is a gateway that provides connectivity to the Internet. The remaining wireless routers form the multi-hop wireless backhaul tier and communicate with each other to eventually forward uplink (resp. downlink) user traffic to (resp. from) the gateway. We model the backhaul mesh routers as an undirected graph G = (R, L), where R = r1 , ..., rn is the set of n routers and L is the set of undirected communication links. An edge (i, j ) ∈ G is referred to as a link, and implies that the mesh routers i and j can communicate with each other as long as both routers share a same radio channel and that both are within radio communication range of each other. For simplicity, we do not integrate in this paper the channel assignment issue. Thus, we assume that each Mesh Router has as many interfaces and orthogonal radio channels as direct mesh router neighbors in graph G. Motivated by the observation that both Gateway and Mesh APs are stationary and are expected to have slow-varying physical channel conditions, we assume the capacity of link e ∈ G, noted C (e), as constant. Without loss of generality, and at the aim of simplicity and the ease of mathematically formalizing our problem, we assume that ∀e ∈ G, C (e) is a multiple of capacity units. B. Paper contributions Since each mesh router operates as a relay node, on the backhaul tier, and as an access point to its attached users, then every link e ∈ G has to support the load generated by its own vertices at the access level as well as the relayed traffic of the backhaul level. Without a careful to capacity assignment at the backhaul level, it is possible that links close to the gateway will be saturated by traffics generated by nearby mesh routers, excluding by this way more distant mesh routers form any access to the gateway. In order to avoid such situation, we propose to assign the capacity of every link e ∈ G, so that: 1) The access to the gateway is guaranteed for every node. 2) Depending on the load requested at the access level, the overall capacity of the backhaul level is max-min fairly shared among the mesh routers at the backhaul level.

3) Without exceeding the capacity of the system, the traffic load at the backhaul level is maximized. In order to fullfil the third requirement, we propose in this paper to consider multipath routing at the backhaul level. The first two requirements can be satisfied when all link capacities are judicously shared, depending on routing paths and bandwidth demands of carried traffics. We formalized this problem as a Maximum Concurrent Flow with Weighted MaxMin Fairness (MCF-WMMF) problem. The formalization and the resolution are detailed in sections IV and V. The needed load at the access level of every mesh router has to be provided as an input for the MCF-WMMF problem. In this work, we propose to estimate this load assuming mobile users that can generate two types of traffic: Voice and Data. In order to support these traffics, and given the elastic nature of data flows, the capacity of the access level has to be assigned according to the bandwidth requirments of each type of traffic. In this paper, we propose two bandwidth sharing strategies, which aim to statistically guarantee that any arriving request (Voice or Data, new call or handover) will not be blocked with a probability above a certain threshold because of the lack of radio ressources at the access level. This problem is formalized in the next section. The required bandwidth capacities at the access level are then used as input for the MCF-WMMF problem. III. C APACITY ASSIGNMENT AT THE ACCESS TIER A. Traffics modeling The access tier is composed of the set of adjacent Mesh AP coverage area. In the following we will denote with cell, the mesh AP coverage area as illustrated in figure 1. Let S be the set of AP cells of the system. As a mobile user may traverse several cells during a communication session, we will distinguish two type of connection requests: (1) new calls (nc) entering to the system, and (2) handoff (ho) demands associated to ongoing connections of customers moving from a cell to a neighbor one. Similarly to the backhaul tier, we also suppose that the total offered bandwidth capacity of a cell i at the access level is a fixed number of bandwidth units, which has to be shared between different users, depending on their QoS requirements. Precisely, we consider two classes of traffics. The first class is related to delay-sensitive applications, like voice, gaming or Video-streaming. For simplicity, we will consider in this paper only voice traffic, which we will refer to it as traffic class 1. To fulfill its QoS requirements, this class requires the allocation of a fixed number of bandwidth units, denoted s1 . The second class is associated to non real-time data traffics, like web browsing, emails or files transfer. We will refer to this data traffic as class 2. Since the associated applications generate variable bit rate traffic, then the allocated bandwidth capacity could be variable over the time. Nevertheless, in order to guarantee the QoS, we consider that a minimum number of capacity units, noted s2 , is required to satisfy any corresponding request. We define T = {1, 2}, then we assume

that the new call requests of a given class j ∈ T arrive to the system according to a Poisson process with a rate λnc ij . Any request of class j ∈ T in the cell i could be rejected if the system does not have enough resources. Otherwise, it is accepted. The connection holding time is supposed to be −1 . We assume that exponentially distributed with mean [μh j] the sojourn duration of a user in a cell i is exponentially −1 . A connection of class j ∈ T distributed with mean [μs i] can leave an area i either the call is terminated or the mobile moves to an adjacent cell (i.e handoff). Hence, the duration time of a class j ∈ T connection in a cell i is exponentially h distributed with mean [μij ]−1 , where μij = μs i + μj . Finally, we suppose that handover requests of class j ∈ T arrive to any cell i according to a Poisson process. Let λho ij and Λho ij be the arrival and the departure handoff rates associated to traffic class j in the cell i, respectively. One can easily see that there is a strong dependency between these rates. Precisely, in one hand, the arrival rate in a given area is dependent on the departure rate from neighbor areas. Formally, λho ij =
l∈Ni

use of each traffic class. In our study, we consider two PC strategies: Strategy 1: The capacity Ci of a cell i ∈ S is divided into two partitions, named Pi1 and Pi2 . ∀i ∈ S and ∀j ∈ T , partition Pij is strictly reserved to traffic class j and has a bandwidth capacity noted Cij . Formally, ∀i ∈ S , Ci = Ci1 + Ci2 . A voice request (new call or handover) arriving to cell i is accepted only if the available bandwidth in Pi1 partition is greater or equal to s1 . Similarly, a data request (new call or handover) arriving to cell i is accepted only if the available bandwidth in Pi2 parition is greater or equal to s2 . The allocated capacity to each class 2 connection in the cell i, noted si2 , will vary according to ongoing class 2 calls in the cell i. Formally, if we note n2 the number of onging data connections in a cell i then: si2 = max(s2 , Ci2 ) and si1 = s1 n2 (3)

Λho lj tli

(1)

where, Ni is the set of neighbor cells to i and tli is the transition probability from the cell i to the neighbor cell l ∈ Ni . On the other hand, the departure handoff rate for class j connections from a cell i can be computed as follow: Λho ij = μs i nc ho + λho (λnc 1 − Bij ij 1 − Bij ) μij ij (2)

nc ho where, Bij and Bij are the nc and ho blocking probabilities of class j ∈ T calls in the cell i ∈ S , respectively.

B. Bandwidth sharing strategies at the access tier The offered capacity of each cell at the access tier, noted Ci , has to be carefully shared so that any connection could be established and maintained with satisfactory QoS requirements. Precisely, the bandwidth sharing strategy has to guarantee the following constraints: Problem 1 QoS problem
nc nc Bij ≤ βj , ho ho Bij ≤ βj

Strategy 2: This strategy is quite similar to the previous one, except that we will distinguish here new calls from handover requests. Precisely, ∀i ∈ S and ∀j ∈ T each partition Pij nc ho is splitted into two disjoint trunks, noted Pij and Pij , with nc ho nc respective capacities Cij and Cij . ∀j ∈ T , partitions Pj and Pjnc are exclusively allocated to nc and ho class j connections, nc ho + Cij . respectively. Formally, ∀i ∈ S and ∀j ∈ T , Cij = Cij A new call (resp. handover) voice request arriving to cell i is accepted only if the available bandwidth in the partition Pinc 1 (resp. Piho 1 ) is greater or equal to s1 . Similarly, a new call (resp. handover) data request arriving to cell i is accepted only if the ho available bandwidth in Pinc 2 (resp. Pi2 ), is greater or equal to nc ho s2 . In a cell i ∈ S , capacities Ci2 and Ci 2 are respectively fairly shared among the ongoing new calls and handover class 2 connections. Therefore, the allocated capacity to each class 2 connection in the cell i, will vary according to ongoing class 2 calls in the cell i and will depends on their nature: new call or handover. C. Dimensioning the partitions size at the access tier Given our modelling assumptions, it is easy to see that each state of the system can be described by a state vector giving for each cell i ∈ S and for each traffic j ∈ T the number of ongoing nc and ho traffic j sessions. That is, the size of each state would be equal to 4n. Since, numerical resolution of such system is intractable, even for small number of cells (n) we will seek in this paper to derive the performances of the system analytically. Precisely, motivated by our Complete Partitioning strategies and given our model assumptions, the evolution of each cell i can be modeled as an Erlang B queue model. Precisely, for strategy 1, ∀i ∈ S and ∀j ∈ T , call blocking probabilities can be expressed as:
nc ho = Bij Bij
ij s 1 ( μij ) ij = Cij Gij !

∀i ∈ S, ∀j ∈ T

nc ho Here, βj and βj are given. nc and ho are tolerated blocking thresholds, respectively. These thresholds are system parameters, which have to be set by the network operator. Among the bandwidth sharing strategies which can be considered, we have interested to the coordinate convex category. This choice is motivated by the fact that the blocking probabilities can be derived analytically, avoiding by this way the difficulties raised by numerical computations. There are several policies which belong to the coordinate convex category. In this paper, we choose the Complete Partitioning (PC) strategy, which is known by its ability to guarantee the QoS constraints of each traffic class. This is simply achieved by allocating a specific bandwidth partition to the exclusive

λ

Cij

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sij

ho where λij = λnc ij + λij and Gij is the normalization constant. Similarly, for strategy 2, ∀i ∈ S , ∀j ∈ T and

∀k ∈ {nc, ho} call blocking probabilities can be expressed as: k
ij s 1 ( μij ) ij k Bij = k k Cij Gij !

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Equations 5 and 4 show clearly that the call blocking probabilities depend on the partitions size. Therefore, solving our constrained problem 1 leads to find ∀i ∈ S and ∀j ∈ T the smallest required partitions size Cij associated to Strategy 1 nc ho and Cij when considering Strategy 2. and Cij 1) Partitions size for Strategy 1: The constrained problem nc ho = Bij ≤ 1 leads to satisfy the following constraints: Bij nc ho min(βj , βj ). The smallest required partitions size can be derived using an incremental approach. Precisely, one can simply set Cij = 1 and then increment the capacities Cij until that the computed nc and ho blocking probabilities, nc ho , βj ). given by equation 4 are below the threshold min(βj Here, the main difficulty raises from the fact that the call blocking probabilities are function of λij , which are dependant on the handoff arrival rates λho ij that are not initially known. To determine these rates we simply use the fixed point algorithm. Precisely, for ∀i ∈ S and ∀j ∈ T and given a chosen Cij value (initially, Cij = 1) we derive an initial estimation of the departure rates using equation 2 and assuming that λho ij = 0. are updated using equation 1. The resulting values Then, λho ij are again injected in equation 2, ∀i ∈ S and ∀j ∈ T . The execution loop eq.2 → eq.1 → eq.2 are repeated ∀i ∈ S and ∀j ∈ T until that λho ij converge toward stable values. The latter values are then injected in equation 4. If, for a given cell j the obtained blocking probabilities are above the threshold then Cij is incremented by one unit and the fixed point algorithm is executed again, assuming the lattest values of λho ij . 2) Partitions size for Strategy 2: Finding the smallest required partitions size for nc calls is trivial using equation nc (∀i ∈ S and 5. One can simply increment the capacity Cij ∀j ∈ T ) until that the resulting nc blocking probabilities, given ho . This algorithm can by equation 5 are below the thresholds βi also be used for the case of ho calls. However, as indicated in the previous section the handoff arrival rates λho ij are unknow at the initialization stage and have to be computed recursively using the fixed point algorithm, similarly to the methodology described in the previous section. IV. C APACITY ASSIGNMENT AT THE BACKHAUL TIER In order to support the cells’ required bandwidths we propose in this paper to exploit the rich connectivity of WMNs to improve the network performances. This can be achieved by maximizing the overall flows in the network by considering more than one path between the sender and the receiver. In this case, the maximization of the end-to-end throughput can leads to adopt the Maximum Concurrent Flow Problem (MCFP) formulation, where all pairs of entities can send and receive flow concurrently [10]. The MCFP is defined as finding the flows fc corresponding to a commodity c, specified by the triplet (sc , tc , dc ), where sc is the source of the traffic, tc is

the sink and finally dc is the flow demand. The purpose is to route, for each commodity c, the overall or a fraction of the demand dc from sc to tc , along several paths such that satisfying the maximum possible portion of all demands and meet some network constraints, such as the sum of flows on any edge e ∈ G does not exceed the edge capacity C (e). Unfortunatly, the Maximum Concurrent Flow Problem does not guarantee that all mesh routers will have an access to the gateway, nor that the allocated bandwidth will be fairly shared among all mesh routers. Indeed, it is possible that the solution which maximizes the flows can lead to a situation where flows that traverse short hop distances will receive a large amount of bandwidth while flows traversing congested links will receive few bandwidth units. In order to avoid this situation a fairness criteria has to be introduced into the MCFP. As in [11] we choose to consider the weighted Max-Min fairness criteria. To formalize our Maximum Concurent Flow with Weighted Max-Min Fairness (MCF-WMMF) problem, we consider two sets of commodities: V and D associated to voice and data flows, respectively. The set V (resp. D) is composed of any voice (resp. data) flow commodity described by the triplet (rg , ri , vi ) (resp. (rg , ri , di )) and associated to any mesh router i. When considering strategy 1 as a bandwidth sharing policy at the access tier, vi = Ci1 (resp. di = Ci2 ). Otherwise, under strategy 2, for any mesh router i the set V (resp. D) will distinguish between two voice (resp. data) flow commodities: one associated to new calls and the other one for handover. In nc nc the first case, vi = Ci 1 (resp. di = Ci2 ) while in the second ho ho (resp. d = C ). one, vi = Ci i 1 i2 Two possible primal formulations of the MCFP are presented in [10]. We are intereasted in this paper to the arc-flow formulation. Our problem is formalized below: Problem 2 MCF-WMMF problem maximize subject to
p∈Pc

γ fc (p) = vc fc (p) ≥ γdc
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(6) ∀c ∈ V ∀c ∈ D ∀e ∈ G (7) (8) (9)

fc (p) ≤ C (e)
c∈K p∈Pc

In this MCF-WMMF problem, the objective γ is a strictly positive real value which represents the equal maximum fraction of allocated bandwidth to all data flow commodities. Pc is the set of all paths that belong to the same commodity c. fc (p) denotes the bandwidth assigned to commodity c on the path p ∈ Pc . Finally, K = V ∪ D. The constraints expressed by equation 7 refer to the fact that any voice commodity c must be assigned its fixed required quantity of bandwidth (i.e. vc ). While equation 8 express the fact that any data commodity c will be assigned higher (if γ ≥ 1) or lower (if γ < 1) bandwidth capacity than dc . In this

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Required/obtained BW in function of users interarrival time

Fig. 3. Required/obtained BW per class in function of users interarrival time

equation, the objective γ ensures that the allocated bandwidth will be fairly shared between all data flow commodities. Finally, equation 9 guarantees that ∀e ∈ G the total flow rate over all commodities is at most C (e). The exact solution of this LP problem could be achived using a solver, and at this aim we used Matlab to get respectively the maximum load factor γ and the assigned flow fc (p) to different commodities. V. N UMERICAL R ESULTS In this section, we evaluate both multi-path routing and shortest path routing algorithms, when considering the overall architecture of the WMN (two tiers). The goal is to determine the required bandwidth for the running applications within each mesh AP cell and assess how well the multi-path routing algorithm responds to the demand of mobile users compared to the shortest path routing algorithm. The simulations topology setting is the same as the one described in section II, considering a grid topology of size 4 × 6 (i.e. 24 mesh nodes), where each Mesh node acts as a Mesh AP with mobile users that are within its coverage area to ensure the functionality of the access tier. We assume that the overall system mobile user call arrival rate, λ, is equal to 1. This call arrival rate is assumed uniformly distributed over all the systems mesh APs, which means that the call arrival rate in any mesh AP, λi , is equal to 24λ. Moreover, we assume that the call mean holding time within any mesh AP, μi −1 = 60s. In all our evaluations, we have fixed the bandwidth required for the voice class to to s1 = 64Kbps and the minimum data traffic requirement to s2 = 512Kbps. The latter one corresponds to the data bandwidth when only one mobile user is attached to a mesh AP. Our analysis is performed according to two scenarios correspond to the two resource allocation strategies described in section III-B. For both scenarios corresponding to S 1 and S 2, the call blocking probability constraints were set to nc nc ho nc , β2 ) = 10−2 , 10−1 and ∀j ∈ {1, 2}, βj = 10−1 βj . (β1 Considering the first scenario, we obtain two commodities, generated from the gateway to each destination (APi , i ∈ {2, 3, .., 24}). One first commodity corresponds to the voice

traffic, and a second commodity corresponds to the DATA traffic. In the following we concentrated our analysis only on DATA traffic because the voice class receives always the requested bandwidth. Figure 2 illustrates the required and the obtained DATA traffic bandwidth for each mesh AP, as a function of mobile users inter-arrival time. In this figure we considered only multi-path routing algorithm. Intuitively, we note that the required capacity in each cell varies in a manner directly proportional to mobile users’ arrival rate. Thus, for high mobile user’s arrival rate (i.e. one user per second), it can be observed that the needed bandwidth of each mesh AP is important, which makes the system unable to provide the minimum desired bandwidth. However, when we consider low mobile users’ arrival rate (i.e. inter-arrival time greater or equal to two seconds), the mesh APs demanded bandwidths are satisfied. On the other hand, when we consider shortest path routing algorithm (Figure 3), we can clearly notice that the whole cells do not receive the same allocated bandwidth, compared to the results obtained in the last figure. Indeed, mesh APs 7, 13 and 19 receive much more bandwidth than they demand and all remaining mesh APs receive their minimum required bandwidth only when the mobile users inter-arrival time is equal to eight seconds. In the second scenario we consider four different commodities, all of them are generated from the gateway towards each destination. They are the combination of the nature of traffic (voice or DATA) and the type of users entering the mesh AP (new call or handover). Figure 4 shows the obtained and requested DATA traffic bandwidth for both new calls and handovers, when considering multipath routing algorithm. Here we observe that for mobile user’s inter-arrival time lower or equal than 2 second, the bandwidths required for both new calls and handovers are not satisfied. Indeed, when the arrival rate is high, that leads to a high number of users per mesh AP, which makes the system unable to provide the minimum required bandwidth. However, when we reduce the mobile users’ arrival rate, mesh APs bandwidth requirements are satisfied for both new calls and handovers. On the other hand, as depicted in Figure 5, when considering shortest path routing protocol a great number of cells did not receive their

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bandwidth or differentiating between the bandwidth affected to new callers and new arrivals from the adjacent mesh APs. This index is equal to 1 even when the number of users connected to the mesh AP is very high. However, using the shortest path routing algorithm shows inefficient bandwidth usage leading to unfair bandwidth distribution among commodities. Comparing the curves illustrating the shortest path when distinguishing between the required bandwidths for both new calls and handovers, we note that the fairness factor for handover is much better than the new calls, suggesting a low blocking probability that ensures service continuity when an application is running. The results show the multipath routing always performs the best and allowed a fairness among different flows. VI. C ONCLUSION In this paper we have given a methodology that allows to dimension a two-tier WMN based on practical assumptions and we have illustrated the enhancement of allocated bandwidth to guarantee the required bandwidth of voice application and satisfy the other applications which do not need a constant bandwidth. At this aim we have coupled the capacity estimation and the Multi-commodity Multi-flow formulation. Testing several scenarios, we confirm that multipath routing is the best to meet the need of mobile users that move between the adjacent coverage areas. We have shown the opportunity of this routing protocol over the shortest path routing. R EFERENCES

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Fig. 5. Required/obtained BW per class in function of users interarrival time

required bandwidth even for new calls. The same result was observed for handovers. As in the previous scenario, we can observe that only mesh APs 7, 13 and 19 that receive much more bandwidth than their demands. Finally, to assess the fairness when using the multi-path and shortest path routing algorithms, we compute the Jain’s Fairness Index given in [12]. Figure 6 represents the fairness index achieved by different users’ arrival rates based on the results of the previous scenarios. We see that for multi-path routing algorithm, the fairness index does not varies and is the same when considering either the overall aggregated

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Strategy 1 - DATA Traffic (short-path) Strategy 2 - DATA Traffic - New Call (short-path) Strategy 2 - DATA Traffic - Handoff (short-path) Strategy 1 - DATA Traffic (multi-path) Strategy 2 - DATA Traffic - New Call (multi-path) Strategy 2 - DATA Traffic - Handoff (multi-path)

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The fairness index for each routing protocol

[1] F. Li, Y. Wang, and X.-Y. Li, “Gateway placement for throughput optimization in wireless mesh networks,” in IEEE ICC’07, pp. 4955– 4960. [2] S. Sen and B. Raman, “Long distance wireless mesh network planning: problem formulation and solution,” in WWW ’07. New York, NY, USA: ACM, pp. 893–902. [3] C. Franklin, A.A. Murthy, “Node placement algorithm for deployment of two-tier wireless mesh networks,” in GLOBECOM ’07. [4] P. Lassila, A. Penttinen, and J. Virtamo, “Dimensioning of wireless mesh networks with flow-level qos requirements,” in PE-WASUN ’06, pp. 17– 24. [5] C. E. Koksal, “Quality-aware routing metrics in wireless mesh networks,,” in Wireless Mesh Networks: Architectures and Protocols. Springer, 2007. [6] A. Raniwala, K. Gopalan, and T. Chiueh, “Centralized channel assignment and routing algorithms for multi-channel wireless mesh networks,” ACM Mobile Computing and Communications Review, vol. 8, pp. 50–65, 2004. [7] M. Alicherry, R. Bhatia, and L. E. Li, “Joint channel assignment and routing for throughput optimization in multi-radio wireless mesh networks,” in MobiCom ’05: Proceedings of the 11th annual international conference on Mobile computing and networking. New York, NY, USA: ACM, 2005, pp. 58–72. [8] E. B.-R. I. Sheriff, “Multipath routing in wireless mesh networks,” Broadnets, 2006. [9] M. Mosko, “Multipath routing in wireless mesh networks,” in in Proc. IEEE Workshop on Wireless Mesh Networks (WiMesh, 2005. [10] F. Shahrokhi and D. W. Matula, “The maximum concurrent flow problem,” J. ACM, vol. 37, no. 2, 1990. [11] M. Allalouf and Y. Shavitt, “Centralized and distributed algorithms for routing and weighted max-min fair bandwidth allocation,” IEEE/ACM Trans. Netw., vol. 16, no. 5, pp. 1015–1024, 2008. [12] R. K. Jain, D.-M. W. Chiu, and W. R. Hawe, “A quantitative measure of fairness and discrimination for resource allocation in shared computer systems,” Tech. Rep., 1984.

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