On the Placement of Internet Taps in Wireless Neighborhood Networks

Transcription

1 1 On the Placement of Internet Taps in Wireless Neighborhoo Networks Lili Qiu, Ranveer Chanra, Kamal Jain, Mohamma Mahian Abstract Recently there has emerge a novel application of wireless technology that enables home users to connect to the Internet by creating a multi-hop wireless network over a neighborhoo region. In such a network, a small number of Internet TAPs (ITAPs) are eploye across the neighborhoo, serving as gateways to the Internet; houses are equippe with low-cost antennas, an form a multi-hop wireless network among themselves to cooperatively route traffic to the Internet using the ITAPs. A similar application also exists in sensor networks, where sensors collect measurement ata an sen it through a multi-hop wireless network to the servers on the Internet via ITAPs. For both applications, placement of ITAPs is a critical eterminant of system performance an resource usage. However there has been little work on this subject. In this paper, we explore the ITAP placement problem uner three wireless link moels. For each link moel, we evelop placement algorithms to make informe placement ecisions base on neighborhoo layouts, user emans, an wireless link characteristics. We also exten our algorithms to provie fault tolerance an to hanle significant workloa variation. We evaluate our placement algorithms through both analysis an simulation. Our results show that our algorithms yiel close to optimal solutions over a wie range of scenarios we have consiere. Keywors: Graph theory, Simulations, Wireless. I. INTRODUCTION Unpreceente growth in wireless technology has mae a tremenous impact on how we communicate. For example, ubiquitous Internet access through wireless has become a reality in many public places, such as airports, malls, coffee shops, hotels, etc. More recently, early aopters have applie wireless technology to obtain broaban access at home, an a number of neighborhoo networks have alreay been launche across the worl [2], [8]. Using wireless as the first mile towars the Internet has a big avantage fast an easy eployment. Therefore it is especially appealing to homes that are out of reach of cable & DSL coverage, such as rural an suburban areas. Even for those areas with cable or DSL coverage, proviing an alternative for Internet access is certainly useful, as it helps to increase network banwith, an suits the iverse nees of ifferent applications. A similar problem of efficiently briging a multi-hop wireless network with the Internet also arises in sensor networks, where sensors collect ata an sen it through a multi-hop wireless network to servers on the Internet via Internet TAPs (ITAPs). Both of the above applications of wireless networks require careful placement of Internet TAPs to enable goo connectivity to the Internet an efficient resource usage. Motivate by these applications, in this paper we explore efficient integration of multi-hop wireless networks with the Internet by placing ITAPs at strategic locations. It shoul be note that an alternative to using multi-hop wireless networks is a cellular approach, which sets up an ITAP to service users within its communication range. However this approach requires significantly more ITAPs than the multi-hop approach, as shown in a previous work [3], an is also ifficult to implement [8]. Therefore, in this paper we focus on the multi-hop approach. Neighborhoo networks have a number of important esign requirements. Such networks shoul be easy an cheap to eploy. Meanwhile they shoul be able to provie Quality of Service (QoS) guarantees to en users. Placement of ITAPs is critical to achieving goo user performance an efficient resource usage in such networks. The esirable properties of placement algorithms inclue (i) efficiently using wireless capacity, (ii) taking into account the impact of wireless interference on network throughput, (iii) robustness in face of failures an changes in user emans. There has been little previous work on this subject. In this paper, we start by formulating the ITAP placement problems uner three wireless moels. For each moel, we evelop algorithms to efficiently place ITAPs in the networks. Our algorithms aim to minimize the number of require ITAPs while guaranteeing users banwith requirements. Next we present a fault tolerance version of the placement algorithm that provies banwith guarantees in the presence of failures. Finally we exten the algorithms to take into account variable traffic emans by eveloping an approximation algorithm to simultaneously optimize ITAP placement base on emans over multiple perios. The rest of this paper is organize as follows. In Section II, we overview relate work. In Section III we escribe the ITAP placement problem an our network moels. In Section IV, we propose a technique to significantly reuce the search space of the potential ITAP locations. In Section V an Section VI, we formulate the placement problems, escribe our algorithms, an evaluate their performance for three wireless link moels. We further valiate these link moels using packet-level simulations in Section VII. In Section VIII, we introuce a fault tolerance version of the placement algorithms an present simulation results. In Section I, we exten our algorithms to hanle variable traffic emans, an evaluate their performance through analysis an simulation. We conclue in Section. II. RELATED WORK There has been a recent surge of interest in builing wireless neighborhoo networks. Some commercial networks that provie Internet access to home users using this technology are escribe in [2], [8]. [2] presents a scheme to buil neighborhoo networks using stanar 82.11b Wi-Fi technology [23] by carefully positioning access points in the community. Such a scheme requires a large number of access points, an irect communication between machines an the access points. This constraint is ifficult to meet in real terrains. The other approach to builing neighborhoo networks is Nokia s Rooftop technology, presente in [8]. This scheme provies broaban access to househols using a multi-hop solution that overcomes the shortcomings of [2]. The iea is to use a mesh network moel with each house eploying a raio, as consiere in this paper. This raio solves the ual purpose of connecting to the Internet an also routing packets for neighboring houses [4]. The eployment an management cost of Internet TAPs in such networks is significant, an therefore it is crucial to minimize the require number of ITAPs to provie QoS an fault tolerance guarantees. However, these problems are not aresse in [2], [8].

2 2 There have been a number of interesting stuies on placing servers at strategic locations for better performance an efficient resource utilization in the Internet. For example, the authors in [22], [19], [25] examine placement of Web proxies or server replicas to optimize clients performance; an Jamin et al. [18] examines the placement problem for Internet instrumentation. The previous work on server placement cannot be applie to our context because they optimize localitn absence of link capacity constraints. This may be fine for the Internet, but is not sufficient for wireless networks since wireless links are often the bottlenecks. Moreover, the impact of wireless interference, an consierations of fault tolerance an workloa variation make the ITAP placement problem verfferent from those stuie earlier. It is worth mentioning that the ITAP placement problem can be consiere as a facility location type of problem. Facility location problems have been consiere extensiveln the fiels of operation research an approximation algorithms (e.g., [21], [28]). Approximation algorithms with goo worst case behavior have been propose for ifferent variants of this problem. However, to the best of our knowlege, these results o not concern the case where links have capacities, as consiere in this paper. The presence of link capacity constraints makes our problem more challenging from a theoretical perspective. The work closest to ours is the pioneering work in [3]. It aims to minimize the number of ITAPs for multi-hop neighborhoo networks base on the assumption that ITAPs use a TDMA scheme to provie Internet access to users. However, TDMA is ifficult to implement in multi-hop networks ue to synchronization an channel constraints. Furthermore, a slotte approach coul result in ecrease throughput ue to unuse slots. In comparison, in this paper we look at more general an efficient MAC schemes, such as IEEE Removing the TDMA MAC assumption yiels completelfferent esigns, an increases applicability of the resulting algorithms. In summary, placing ITAPs uner the impacts of link capacity constraints, wireless interference, fault tolerance, an variable traffic emans is a unique challenge that we aim to aress in this paper. III. PROBLEM DESCRIPTION AND NETWORK MODEL The ITAP-placement problem, in its simplest form, is to place a minimum number of ITAPs that can serve a given set of noes on a plane, which we call houses. A house h is sai to be successfully serve, if its eman, w h, is satisfie by the ITAP placement. A house h is serve by an ITAP i through a path between h an i. This path is allowe to pass through other houses, but any two consecutive points on this path must have wireless connectivity between them. We are usuallntereste in the fractional version of this problem. That is, we consier the flexibility that a house is allowe to route its traffic over multiple paths to reach an ITAP. This problem can be moele using the following graphtheoretic scheme. Let H enote the set of houses an I enote the set of possible ITAP positions. We construct a graph G on the set of vertices H[Iby connecting two noes if an only if there is wireless connectivity between them. The goal is to open the smallest number of ITAPs (enote by the set I ), such that in the graph G[H [I ], one can route w h units of traffic from house h to points in I simultaneously, without violating capacity constraints on vertices an eges of the graph, where w h is the eman from house h. The ege capacity, Cap e, in the graph enotes the capacity of a wireless link. In aition, each noe also has an upper boun on how fast traffic can go through it. Therefore, we also assign each noe with a capacity, Cap h. Usually Cap h = Cap e,as both represent the capacity of a wireless link. (Our schemes work even when Cap h 6= Cap e, e.g., when a noe s processing spee becomes the bottleneck.) Moreover, each ITAP also has a capacity limit, base on its connection to the Internet an its processing spee. We call this capacity, the ITAP capacity, Cap i. In aition to ege an vertex capacities an house emans, another input to the placement algorithms is a wireless connectivity graph (among houses). We can etermine whether two houses have wireless connectivity using real measurements, an give the connectivity graph to our placement algorithms for eciing ITAP locations. In our performance evaluation, since we o not have wireless connectivity graphs base on real measurements, we instea erive connectivity graphs base on the protocol moel, introuce in [14]. In this moel, two noes i an j can communicate irectly with each other if an only if their Eucliean istance is within a communication raius, CR. Given the position of all the noes, we can easily construct a connectivity graph by connecting two noes with an ege if their istance is within CR. However our placement algorithms can also work with other wireless connectivity moels (e.g., physical moel [14] or base on real measurements). In the following sections, we stuy several variants of this placement problem. A. Incorporating Wireless Interference There are several possible ways to moel wireless interference. One approach is to use a fine-graine interference moel base on the notion of a conflict graph, introuce in [17]. The conflict graph inicates which groups of links mutuallnterfere an hence cannot be active simultaneously. As shown in [17], the conflict graph moel is flexible to capture a wie variety of wireless technologies, such as irectional antennas, multiple raios per noe, multiple wireless channels, an ifferent MAC protocols. The impact of wireless interference can be expresse as a set of linear constraints [17], an the ITAP placement problem can then be solve bteratively solving linear programs (similar to the approach escribe in Section VI). The main challenge of using the fine-graine interference moel is high complexity (sometimes prohibitive), since for even a moerate-size network the number of interference constraints can become hunres of thousans. An alternative approach is to use a coarse-graine interference moel that captures the tren of throughput egraation ue to wireless interference. Since there are usually a limite number of wireless channels available, not all links can be active at the same time to avoi interference. As a result, wireless throughput generally egraes with the number of hops in the path as we show in the following scenario. Consier a linearchain network, where each link has a unit capacity. Since the interference range of a noe is typically larger than the communication range [27], it is possible that all the noes in the chain interfere with each other. In this case, only one link can be active at a time, which suggests that the maximum throughput from noe to noe n is k n for k<nan 1 for k n, where k is the number of available channels, an n is the number of hops. As we can see, if we have enough channels, the throughput can approach the channel capacity. On the other han, if we only have one channel, then throughput egraes as a function of 1 n. This has also been confirme by several simulation stuies base on an other MAC protocols similar to [16], [13].

3 3 In practice, the network topology can be more complicate, an the relationship between throughput egraation an an increasing hop-count epens on many factors, such as communication vs. interference range, the types of antenna (irectional vs. omni-irectional), MAC protocols, the number of contening raios, etc. There is no single function that can capture the impact of interference on wireless throughput. Therefore, we stuy the placement problem uner several link moels. We escribe the link moels using two relate functions. In our iscussion, T hroughput l enotes the amount of throughput on a link along a path of length l, assuming each wireless link capacity is 1. The other function, g(l), enotes the amount of link capacity consume if it is on a path of length l an the en-to-en 1 throughput of the path is 1. It is clear that g(l) = throughput l, since in orer to get one unit throughput along a path of length 1 l, we nee to have throughput l capacity at each ege along the path, assuming the en-to-en throughput increases proportionally with the ege capacity, which is true in practice. In this paper, we stuy the following moels separately: 1) Ieal link moel: If throughput l =1for all l, or equivalently, g(l) = 1, we get the basic version of the problem. This moel is appropriate for the environment with very efficient use of spectrum. A number of technologies, such as irectional antennas (e.g., [7], [11]), power control, multiple raios an multiple channels, all strive to achieve close to this moel by minimizing throughput egraation ue to wireless interference. 2) General link moel: A more general moel is when throughput l or g(l) is an arbitrary function of l. As we will show in Section VI, we can formulate the ITAP placement problem for the general link moel as an integer linear program, an evelop polynomial placement algorithms. In aition, we also evelop more efficient heuristics for two forms of g(l). a) Boune hop-count moel: If throughput l = 1 for l» k an throughput l = for l > k (or equivalently, g(l) = 1 for l» k an g(l) = 1 for l > k), we get a variant in which flow cannot be route through paths of length more than k. This approximates the case where we try to ensure each flow gets at least a threshol amount of throughput by avoiing paths that excees a hop-count threshol. b) Smooth throughput egraation moel: This correspons to the case when throughput l = 1 l, where l is the number of hops in the path. This is equivalent to g(l) =l for all l s (i.e., the capacity consume is equal to the flow times the number of hops). This represents a conservative estimate on throughput in a linear-chain network as we show above, an therefore this moel is appropriate when tight banwith guarantees are esire. Note that we only moel interference an contention among noes whose paths share common links or noes. A more accurate moel will have to hanle interference among noes on inepenent paths, using schemes such as the conflict graph [17]. However, in Section VII, we valiate our link moels using packet-level simulations, an show that an ITAP placement base on the above moels gives satisfactory performance in practice. B. Incorporating Fault Tolerance Consieration A multi-hop scheme for builing neighborhoo networks has a number of avantages, such as a reuce number of ITAPs, an ease of eployment among others [8]. However, such a scheme also requires ifferent houses along a path to the ITAP to forwar the traffic to an from a house. The banwith requirements of a house may not be satisfie if even one house ecies to shut itself own. Furthermore, ITAPs may be temporarily own. Our placement scheme hanles such scenarios by routing traffic through multiple inepenent paths, an overprovisioning the elivery paths. The fault tolerance consieration has significant impact on the placement ecision. We will present the etails in Section VIII. C. Incorporating Workloa Variation Several stuies show that user traffic emans exhibit iurnal patterns (e.g., [6], [2], [26]). Since it is not easy to change ITAP locations once they are eploye, ieally these ITAPs shoul hanle emans over all perios. In Section I, we present algorithms to simultaneously optimize ITAP locations base on workloa uring ifferent perios. D. Generic Approach In the following sections, we will investigate ifferent variants of the placement problem. Our general approach is as follows. Given a set of potential ITAP locations, which may inclue all or a subset of points in the neighborhoo, we first apply the reuction algorithm escribe in Section IV to prune the search space. Then base on our choice of wireless link moel, fault-tolerance requirement, an eman variation, we choose one of the placement algorithms escribe in Section V through Section I to etermine ITAP locations. IV. REDUCING THE SEARCH SPACE FOR ITAP POSITIONS All points on the plane coul be potential ITAP locations. To make search tractable, we escribe a pruning algorithm. It is base on the following two observations. Equivalence class: First, we can group points on the plane into equivalence classes, where each equivalence class is represente by the set of houses that are reachable via a wireless link. If we allow placing multiple ITAPs at the same location, then it is straightforwar to show that searching over all points on the plane is equivalent to searching over all the equivalence classes (i.e., we only nee to pick one noe from each equivalence class for our search, since all the points in one equivalence class are equivalent as far as ITAP placement is concerne.) In this way, we reuce the number of ITAP locations to EC, where EC is the number of equivalence classes. Pruning: The number of equivalence classes can still be large. We can further prune the search space using the following heuristic. Given two equivalence classes A an B, let houses A an houses B enote the set of houses that have wireless connectivity with classes A an B, respectively. If houses A houses B (i.e., A is covere by B), then we prune the class A. The above reuction schemes help significantly reuce the search space. V. IDEAL LINK MODEL First, we consier the placement problem for the ieal link moel. We formulate the problem as a linear program, an present several placement algorithms an their performance results.

4 4 A. Problem Formulation We formulate the placement problem for the ieal link moel as an integer linear program shown in Figure 1. For each ege e an house h, we have a variable x e;h to inicate the amount of flow from h to ITAPs that is route through e. For each ITAP i we have a variable that inicates the number of ITAPs opene at the location i (More precisely, is the number of ITAPs opene at locations in the equivalence class i, where the equivalence class is introuce in Section III.) Cap e, Cap h, an Cap i enote the capacity of the ege e, house h, an ITAP i, respectively; w h enotes the traffic eman generate from house h. i2i minimize subject to e=(h;v) e=(v;h) h e=(v;h ) x e;h = x e;h w h x e;h = x e;h» Cap e h ;e=(v;h) h ;e=(v;i) e=(v;i) e=(h ;v) x e;h» Cap h x e;h» Cap i x e;h» w h x e;h 8h; h 2H;h 6= h 8h 2H 8h 2H 8e 2 E(G) 8h 2H 8i 2I 8i 2I;h 2H x e;h 8e 2 E(G);h 2H 2f; 1; 2;:::g 8i 2 I Fig. 1. LP formulation for the ieal link moel Now we present a brief explanation of the above integer linear program. The first constraint ( P e=(v;h ) x e;h = Pe=(h ;v) x e;h) formulates the flow conservation constraint, i.e., for every house except the house originating the flow, the total amount of flow entering the house is equal to the total amount of flow exiting it. The inequality P e=(h;v) x e;h w h formulates the constraint that each house has w h amount of flow to sen, an the thir constraint inicates that a house oes not receive flow sent btself. The next three inequalities of the above program capture the capacity constraints on the eges, houses, an ITAPs. The inequality P e=(v;i) x e;h» w h says that no house is allowe to sen any traffic to an ITAP unless the ITAP is open. Notice that this inequalits reunant an follows from the ITAP capacity constraint an the assumption that is an integer. However, if we want to relax the integrality assumption on s in orer to erive a lower boun using an LP solver (see Section V-B.5 for example), then it is important to inclue this inequalitn the linear program so that we can get a tighter lower boun. The following theorem shows that it is computationally har to optimally solve the ITAP placement problem for the ieal link moel. Refer to Appenix for the proof. Theorem 1: It is NP-har to fin a minimum number of ITAPs require to cover a neighborhoo in an ieal link moel. Moreover, the problem has no polynomial approximation algorithm with an approximation ratio better than ln n unless P = NP. B. Placement Algorithms In this section, we escribe various placement algorithms for the ITAP placement problem uner the ieal link moel. In particular we look at the greey, augmenting, clustering an ranom algorithms. 1) Greey Placement: We esign the following greey placement. We iteratively pick an ITAP that maximizes the total emans satisfie when opene in conjunction with the ITAPs chosen in the previous iterations. The major challenge is to etermine how to make a greey move in each iteration. This involves efficiently computing the total user emans that can be serve by a given set of ITAPs. We make an important observation: computing the total satisfie emans can be formulate as a network flow problem. This suggests that we can apply the network flow algorithms [9] to efficiently etermine the satisfie emans. A few transformations are require to make the network flow algorithm applicable. Figure 2 shows a skeleton of the algorithm, which fins a multiset S of ITAPs to open, where a multiset is the same as a set, except that it allows uplicate elements. Allowing uplicate elements in S inicates that we can open multiple ITAPs in the locations that belong to the same equivalence class (i.e., reachable from the same set of houses), which is certainly feasible. Input: Set of houses H, set of ITAPs I, graph G on the set H[Iwith capacities on its eges an vertices. Output: A multiset S of ITAPs to be opene. begin S := ;; F low := ; while Flowis less than the total eman o max := ; for each j 2Io ffl Let G be the subgraph of G inuce on H[S [fjg, with the same capacities as G. (If there are uplicates in S [fjg, we create one point for each uplicate element.) ffl For each house, transform its vertex capacity constraint to an ege capacity constraint by replacing the house h with two noes, in h an out h ; an connect in h to out h using a irecte ege with capacity cap h ; all incoming eges towar the house go to in h an all out-going eges from h come from out h. ffl A two vertices s an t to G, eges of capacity w h from s to each h 2H, an eges of capacity cap i from each i 2 S[fjg to t. ffl Fin the maximum flow from s to t in G ; Let f be the value of this flow. ffl if f>max, then max := f; bestit AP := j; enfor; S := S [fbestit AP g; F low := max; enwhile; en. Fig. 2. Greey placement algorithm in the ieal link moel The following theorem shows a worst-case boun on the performance of the above algorithm. An empirical performance analysis of this algorithm is presente in Section V-C. Theorem 2: Consier the ITAP placement problem in the ieal link moel with integral emans an integral house an link capacities, an let D enote the total eman of the houses. The approximation factor of the greey algorithm for this problem is at most ln(d). In other wors, if the optimal solution for the ITAP placement problem opens K ITAPs, the greey algorithm opens at most K ln(d) ITAPs. We will nee the following lemma to prove the above theorem. This lemma is non-trivial an uses the For-Fulkerson maximum flow-minimum cut theorem [9] in the proof. Lemma 3: Assume a multiset S of ITAPs are opene. Consier an optimal way of routing the maximum total emans from houses to the ITAPs in S, an let f i enote the amount of traffic route to ITAP i in this solution, where i 2 S. Assume that at a later time, a multiset S [T of ITAPs are opene. Then, there is an optimal way of routing the maximum total emans from houses to these ITAPs in which f i units of traffic is route to ITAP i for ever 2 S. Refer to Appenix for the proofs of the above theorem an lemma. Base on Theorem 2, we have the following corollary. Corollary 4: Let N be the number of houses. The approx-

5 5 imation factor of the greey algorithm in the ieal link moel is ln(n ) when the capacities of eges an vertices are integervalue an every house has either zero or one unit of eman. Remark 1. Corollary 4 in combination with Theorem 1, shows that this algorithm achieves the best possible (worst-case) approximation ratio for the graph theoretic moel when every house has either zero or one unit of eman. Furthermore, even though in our moel we allow fractional routing of the flow, our greey algorithm always fins an integral solution in this case, i.e., the eman from each house will be serve through one path to an open ITAP. This is a consequence of the integrality theorem [9]. Remark 2. Notice that ln(d) is the worst-case boun for heterogeneous emans. To make the worst-case boun tighter, we can normalize house emans, ege capacities, an noe capacities before we apply the greey placement algorithm. This yiels a lower approximation factor, since D is reuce after normalization. Moreover, as we will show later in this section, in practice the greey algorithm performs quite close to the optimal, an much better than the worst-case bouns, ln(d) or ln(n ). 2) Augmenting Placement: The iea of the augmenting placement algorithm is similar to the greey algorithm. The main ifference in the augmenting algorithm is that we o not make a greey move; instea we are satisfie with any ITAP that leas to an increase in the amount of supporte flow. More specifically, we search over the set of possible ITAP locations, an open the first ITAP that results in an increase in the amount of flow when opene together with the alreay opene ITAPs. 3) Clustering-base Placement: We compare our placement algorithms to the clustering-base scheme, propose in [3]. The basic iea of the algorithm is to partition the network noes into a minimum number of isjoint clusters, an place an ITAP in each cluster. We use the Greey Dominating Inepenent Set (DIS) [3] heuristic to etermine a set of clusterheas, which are use as possible ITAP locations. The noes are then clustere to ensure that each noe is associate with the closest clusterhea, an a shortest path tree roote at the clusterhea is use for sening packets from an elivering packets to the cluster. The cluster is further ivie into sub-clusters if either the weight or relay-loa constraints are violate. The weight constraint specifies that an ITAP can serve noes as long as the sum of their emans oes not excee the capacity of the ITAP, an the relay-loa constraint specifies an upper boun on the maximum flow that can go through a noe in the neighborhoo cluster. In our simulations we use the ITAP capacitnstea of wireless capacity when checking the weight constraint of placing an ITAP at a particular house; this is necessary since the ITAP capacity can be greater than the wireless capacitn our simulations. This ensures a fair comparison of the clustering algorithm with our placement schemes. We refer the reaer to [3] for more etails of this algorithm. 4) Ranom Placement: This algorithm ranomly places an ITAP at a house iteratively until all the user emans are satisfie. To avoi wasting resource, it ensures that each house has at most one ITAP. This approximates un-coorinate eployment of ITAPs in a neighborhoo, an gives a baseline to evaluate the benefits of the more sophisticate algorithms presente above. 5) Lower Boun: It is useful to compare our algorithms with the optimal solution. However, our problem is NP-har, an we cannot erive an optimal solution. Therefore we compare our algorithms with the lower bouns. We erive the lower boun by relaxing the integer constraints on an solving the relaxe LP problem using cplex [1]. The lower boun is a useful ata point to compare with, as it gives an upper boun on how much a practical algorithm iffers from the optimal. In other wors, if an algorithm s performance iffers from the lower boun by ffi, it means that its ifference from the optimal is at most ffi. C. Performance Evaluation In this section, we evaluate the performance of the above placement algorithms uner various scenarios. We use the following notations in our iscussion. ffl N: the number of houses ffl WC: a wireless link s capacity ffl IC: an ITAP s capacity ffl CR: communication raius ffl HR: average inter-house istance ffl w h : house h s eman We compare the performance of ifferent algorithms by varying each of the above parameters. In our evaluation, we use both ranom topologies an a real neighborhoo topology. The ranom topologies are generate by ranomly placing houses in a region of size N Λ N, an varying the communication raius. The real neighborhoo topology contains 15 houses, spanning over a region of 116m*113m. (We cannot reveal the source of the ata for confientiality.) The average inter-house istance in the real topologs 74 meters. (We etermine the interhouse istance by averaging the istance between a house an its closest neighbor.) Unless otherwise specifie, for the same parameter setting, we run simulations three times, an report the average number of ITAPs require for each placement algorithm. Effects of the communication raius: We start by examining the effect of communication raius (CR) on the placement algorithms. It is easy to see from the problem formulation that CR only the ratio, HR, is important. Therefore in our evaluation, we vary the communication raius from 1 to 5, while fixing the inter-house istance by ranomly placing 1 houses in an area of 1*1, which yiels an average inter-house istance of Figure 3 illustrates the number of ITAPs require on varying CR. We make the following observations. First, we see that an increase in CR results in a greater overlap of wireless coverage of the houses, an therefore fewer ITAPs are sufficient to satisfy the house emans. Secon, comparing the performance across ifferent algorithms, we observe that the greey algorithm performs very close to the lower boun over all cases. Interestingly, the augmenting algorithm performs quite well, too. The goo performance of the augmenting algorithm comes from the requirement that new ITAPs shoul lea to throughput improvement, which avois wasting resource on the alreay covere region. This is especially useful after several ITAPs have been place, since at this point only a few locations remain that can further increase the satisfie emans. In contrast, the clustering an ranom-house placement schemes perform much worse. Compare to the greey strategy, both schemes often require 2 to 1 times as many ITAPs. Note that when the communication raius is very large, the clustering algorithm yiels worse performance than the ranom-house placement. This is because in the clustering algorithm ata issemination follows a shortest path tree, instea of maximizing the amount of flow that can be pushe to the ITAPs. In comparison, the other algorithms, incluing the ranom-house placement, run the network algorithm to maximize the total satisfie emans. Effects of network size: Next we stuy the impact of network size on the placement algorithms. We ranomly place N

6 communication raius greey augment cluster ranom lower boun Fig. 3. Ieal link moel: varying communication raius, where N = 1, WC =6, IC =1, an w h =18h 2 H. houses in an N Λ N area while fixing the communication raius to 1. Figure 4 shows the number of require ITAPs using the ifferent placement algorithms for various network sizes. As we woul expect, an increase in the number of houses leas to a larger number of ITAPs require to cover the neighborhoo. Moreover, the greey algorithm continues to perform very well, with its curve mostly overlapping with the lower boun. The augmenting algorithm performs slightly worse, whereas the clustering an ranom algorithms perform much worse requiring up to 5 an 8 times as many ITAPs, respectively. In aition, the benefit of greey algorithm increases as the network gets larger # houses greey augment cluster ranom lower boun Fig. 4. Ieal link moel: varying the number of noes, where CR= 1, WC = 6, IC =1, an w h =18h 2 H. Effects of wireless link capacity: We also stuy the effects of wireless banwith on the placement algorithms. As shown in Figure 5, the relative ranking of the algorithms stays the same. The effect of banwith is only pronounce when it is very limite. For example, when the wireless banwith is equal to a single house s eman, the number of ITAPs require is consierably large. As the banwith increases an the wireless link is no longer the bottleneck, the number of require ITAPs remains the same with a further increase in the wireless link capacity Wireless capacity greey augment cluster ranom lower boun Fig. 5. Ieal link moel: varying wireless link capacity, where N = 1, CR =1, IC =1, an w h =18h 2 H. Effects of the ITAP capacity: We compare the placement algorithms by varying the ITAP capacity. As Figure 6 shows, when ITAP capacits small an hence is a bottleneck, the number of require ITAPs ecreases proportionally with an increase in ITAP capacity. As the ITAP capacits large enough an no longer the bottleneck, the number of require ITAPs is unaffecte by a further increase in ITAP capacity. Moreover, the relative performance of ifferent placement algorithms is consistent with the previous scenarios ITAP capacity greey augment cluster ranom lower boun Fig. 6. Ieal link moel: varying ITAP capacity, where N = 1, CR = 1, WC =6, an w h =18h 2 H. Effects of heterogeneous house emans: Sofarwehave consiere homogeneous house emans (i.e., each house generates one unit eman). A number of stuies show that realistic user emans are very heterogeneous, an often exhibit Zipf-like istributions [5], [6]. Motivate by these finings, below we evaluate the placement algorithms when house emans follow a Zipf istribution. Figure 7 summarizes our results. As it shows, the results are qualitatively the same as those of using the homogeneous house emans. The greey algorithm continues to out-perform the others significantly, an yiel nearly optimal solutions # houses greey augment cluster ranom lower boun Fig. 7. Ieal link moel: varying the number of noes, where CR = 1, WC =6, IC =1, an the house emans follow a Zipf-istribution. Real neighborhoo topology: Finally we evaluate the placement algorithms using a real neighborhoo topology of 15 houses. We again use Zipf-istribute house emans. As shown in Figure 8, initially when the communication range is too small, most houses are unreachable from other houses, an therefore all the algorithms require close to 15 ITAPs. As the communication range increases, fewer ITAPs are neee to cover the neighborhoo. At the extreme, when the communication range reaches 25 meters, the neighborhoo forms a single connecte component, an therefore most algorithms require only one ITAP. (Note that this is only true for the ieal moel. As shown in the next section, when consiering wireless interference, we often nees more ITAPs even for a single connecte component.) Moreover, the greey algorithm performs close to optimal over all communication raii consiere.

7 communication raius (meters) greey augment clustering ranom lower boun Fig. 8. Ieal link moel: results of a real neighborhoo topology with various communication raii, where N = 15, WC =6, IC = 1, an the house emans follow a Zipf istribution. VI. GENERAL LINK MODEL The problem of efficient ITAP placement is more challenging when the throughput along a path varies with the path length. This correspons to the general link moel introuce in Section III-A. In this section, we first formulate the problem for a link moel with an arbitrary throughput egraation function, an then present efficient heuristics for two variants of this egraation function. A. Problem Formulation We formulate the placement problem for the general link moel as an integer linear program shown in Figure 9. In this program x e;h;l;j enotes the total amount of flow route from house h to the ITAPs using a path of length l when ege e is the j th ege along the path. Variable is an inicator of the number of ITAPs opene in the equivalence class i, an each house h has w h units of traffic to sen. The throughput egraation function for a path of length l is enote by g(l). L is an upper boun on the number of hops on a communication path, an if there is no such upper boun, we set L = jhj. The other variables in the program are similar to the ones use by the program presente in Figure 1. i2i minimize subject to e=(v;h ) e=(h;v);l h;l;j»l x e;h;l;j = x e;h;l;1 w h e=(h ;v) g(l) x e;h;l;j» Cap e h ;e=(v;h);l;j»l h ;e=(v;i);l;j»l e=(u;i);l;j»l g(l) x e;h ;l;j» Cap h g(l) x e;h ;l;j» Cap i x e;h;l;j» w h x e;h;l;j+1 8h; h 2 H; h 6= h; l; j 2f1;:::;Lg;j < l 8h 2 H 8e 2 E(G) 8h 2 H 8i 2 I 8i 2 I;h 2 H x e;h;l;j 8e 2 E(G);h 2 H; 2f; 1; 2;:::g l; j 2f1;:::;Lg;j» l 8i 2 I Fig. 9. LP formulation for the general link moel, where g(l) moels throughput egraation with increasing hop-count. The following theorem is an immeiate consequence of Theorem 1, as the ieal link moel is a special case of the general link moel, when g(l) =1. Theorem 5: It is NP-har to fin a minimum number of ITAPs to cover a neighborhoo for a general link moel. B. Placement Algorithms In this section, we present placement algorithms for the general wireless link moels. First we look at a general throughput egraation moel, an then escribe more efficient algorithms for two special cases: boune hop-count an smooth throughput egraation. 1) Greey Algorithm: The high-level iea of the greey algorithm is similar to the one presente for the ieal link moel. We iteratively select ITAPs to maximize the total user emans satisfie. The new challenge is to etermine a greey move in this moel. Unlike in the ieal link moel, we cannot compute the total satisfie emans by moeling it as a network flow problem since the amount of flow now epens on the path length. As we will escribe below, this computation can be one by solving a linear program, or by using a heuristic. Expensive algorithm for the general link moel: Without making assumptions about g(l), we can compute the total satisfie user emans, for a given set I of ITAPs, by solving a slightly moifie LP problem than the one in Figure 9. In this linear program, we replace the variable by the number of occurrences of i in I (This amounts to removing all the variables corresponing to eges ening in ITAP positions outsie I an removing inequalities containing these variables). The objective will be to maximize P hpe=(h;v);l x e;h;l;1, which correspons to maximizing the satisfie emans. We also moify the secon constraint to be P e=(h;v);l x e;h;l;1» w h in orer to limit the maximum flow from each house h. In theory, solving a linear program takes polynomial time. However, in practice an LP solver, such as cplex [1], can only hanle small-size networks uner this moel ue to the fast increase in the number of variables an constraints with the network size. Below we consier two forms of g(l): (i) boune hopcount: g(l) =1for all l» k, an g(l) =1 for l > k, an (ii) smooth egraation: g l = l for all l. We evelop more efficient greey algorithms for both cases. Efficient algorithm for the boune hop-count moel: We can use the following greey algorithm to fin the total emans satisfie by a given set of ITAPs. In each iteration, the algorithm fins the shortest path from eman points to opene ITAPs in the resiual graph, routes one unit of flow along this path, an ecrease the capacities of the eges on the path by one in the resiual graph. This is continue until the shortest path foun has length more than the hop-count boun. This algorithm is similar to the algorithm propose in [15] for a similar problem. While this heuristic oes not guarantee computing the maximum flow (so each greey step is not local optimal), it works very well in practice as shown in Section VI-C.1. Efficient algorithm for the smooth throughput egraation moel: When P g(l) = l or throughput l = 1 l, the total emans satisfie by a set of ITAPs are given by the expression: maximize p 1 i2p jp ij where P is a collection of egeisjoint paths in the graph, an jp i j enotes the length of the path p i. Therefore to maximize this objective function, our heuristic shoul prefer imbalance in path lengths, an this motivates the algorithm we escribe below. As the heuristic for the boune hop-count moel, in the smooth throughput egraation moel we compute the total satisfie emans by the selecte ITAPs through iteratively removing shortest paths in the resiual graph. However, we make the following moifications. First we continue picking paths until there is no path between any eman point an any open ITAP. Secon, after we obtain all the paths, the eman satisfie along each path p, enote as SD p, is compute accoring

8 8 to the throughput function, throughput(l) =1=l, an the total satisfie emans are the sum of SD p over all paths p. Although this algorithm oes not always fin the maximum flow (so each greey step is not local optimal), it yiels very goo performance as shown in Section VI-C.2. 2) Augmenting Algorithm: We use the same algorithm escribe in Section VI-B.1 to compute the total emans satisfie by a given set of ITAPs. The ifference between the greey an augmenting algorithms is in the way ITAPs are selecte in each iteration. While the greey algorithm selects an ITAP that results in a maximum increase in the supporte emans, the augmenting algorithm picks the first ITAP that leas to an increase in the supporte emans. 3) Ranom Algorithm: The ranom algorithm ranomly picks a house to eploy an ITAP until all the emans are satisfie. Again this approximates uncoorinate ITAP eployment. We use the algorithm escribe in Section VI-B.1 to etermine the total emans that can be supporte by the selecte ITAPs in each iteration. 4) Clustering-base Placement: We also apply the clustering-base placement in [3] to the boune-hop count moel with the following moification. We ivie a cluster into sub-clusters not only when the weight or relay-loa constraints are violate, but also when the istance between any noe an its clusterhea excees the hop-count threshol. The algorithm, however, oes not apply to the smooth throughput egraation moel. 5) Lower Boun: The lower boun can be erive by relaxing the integrality constraint in Figure 9, an solving the relaxe linear program. However, although in theory linear programs can be solve in polynomial time, we were unable to solve the program in Figure 9 for large networks, ue to the memory constraints. So in our performance evaluation section, we use the solution to the LP formulation of Figure 1 for the ieal link moel, as the lower boun for the general case too. This lower boun is always correct, since the ieal link moel is a relaxation of the general moel. However, it might not be tight, since it ignores the throughput egraation with hop count, an therefore requires fewer ITAPs. However, we show in Section VI-C that the results from our greey an augmenting algorithms are still close to these loose lower bouns. C. Performance Evaluation In this section, we evaluate the performance of placement algorithms for boune-hop count an smooth throughput egraation moels. 1) Boune Hop-count Moel: We compare the placement algorithms for boune-hop count moel by varying the hop-count threshol, communication raius, an neighborhoo topology. Effects of hop-count threshol: First we compare the placement algorithms by varying the hop-count threshol. As shown in Figure 1, when the hop-count threshol increases, the effect of hop-count reuces, since all or most paths are within hopcount limit. Comparing the ifferent placement algorithms, we see that the greey placement performs very close to the lower boun, especially for large hop-count threshols. When the hop-count threshol is small, the gap between the lower boun an greey algorithm is slightly larger, since the lower boun ignores throughput egraation with the hop-count, an is not as tight. Compare to the greey algorithm, the augmenting algorithm requires 5% more ITAPs; the clustering algorithm in [3], requires 2-3 times as many ITAPs; an the ranom algorithm requires 4 to 8 times as many ITAPs hop-count threshol greey augment cluster ranom lower boun Fig. 1. Boune hop-count moel: varying the hop-count threshol, where N =1, CR =1, WC =6, IC = 1, an w h =18h 2 H. Effects of communication raius: Next we fix the hopcount threshol to 3, an vary the communication raius. As Figure 11 shows, an increase in communication raius reuces the number of ITAPs require to cover the neighborhoo. Moreover the greey continues to perform significantly better than the alternatives. We observe similar results for other hopcount threshols. hop-count threshol = communication raius greey augment cluster ranom lower boun Fig. 11. Boune hop-count moel: varying the communication raius, where N = 1, WC = 6, IC = 1, hop-count threshol = 3, an w h = 1 8h 2 H. Real neighborhoo topology: We also evaluate the placement algorithms using the real neighborhoo topology. As shown in Figure 12, the results are qualitatively the same as the ranom topologies. The greey algorithm performs very close to the lower boun for all the communication raii consiere hop-count threshol= communication raius (meters) greey augment clustering ranom lower boun Fig. 12. Boune hop-count moel: results of a real neighborhoo topology for various communication raii, where N = 15, WC = 6, IC = 1, hop-count threshol = 3, an the house emans follow a Zipf istribution. 2) Smooth Throughput Degraation Moel: Next we empirically stuy the placement algorithms for the smooth throughput egraation moel. Effects of communication raius: Figure 13 compares the performance of ifferent placement algorithms for the smooth throughput egraation link moel on varying the communication raius. As we can see, the number of require ITAPs

9 9 ecreases as the communication raius increases. The gap between ifferent algorithms performance is the largest when the communication raius is between 5 an 2 (The average interhouse is aroun 5.). This occurs because when the raius is very small, most houses are isconnecte from one another, an therefore the number of ITAPs require is close to the number of houses regarless of placement algorithms; when the raius is very large, most houses are reachable from one another within one or few hops, an the number of ITAPs require becomes close to 1. In comparison, for meium communication raius, which is the most likely scenario in practice, the gap between the ifferent algorithms is most significant. This is especially so when we compare the ranom placement with the other two. Note that the lower boun, which is erive bgnoring the impact of hop-count on throughput, is more loose for this scenario. Nevertheless the grees still competitive when compare with these loose lower bouns communication raius greey augment ranom lower boun Fig. 13. Smooth throughput egraation moel: varying the communication raius, where N =1, WC =6, IC =1, an w h =18h 2 H. Real neighborhoo topology: Figure 14 shows the results from the real neighborhoo topology. As we can see, the greey place continues to perform well, yieling close to optimal performance communication raius (meters) greey augment ranom lower boun Fig. 14. Smooth throughput egraation moel: results of a real neighborhoo topology for various communication raii, where N =15, WC =6, IC = 1, an the house emans follow a Zipf istribution. VII. VALIDATION To valiate the wireless link moels use in this paper, we run simulations in Qualnet [1], a commercial network simulator. More specifically, given a neighborhoo layout, the placement algorithms etermine the ITAP locations an the set of paths each house uses to reach the ITAPs. We use the same neighborhoo layout an ITAP locations in the simulations. Every noe in the simulation uses an omni-irectional antenna an 82.11b MAC, with the communication range an interference range being 195 meters an 376 meters, respectively. Every house sens CBR traffic to the ITAPs at the rate specifie by the placement algorithms output. To support multi-path routing, we implemente probabilistic source-routing in Qualnet, where the paths use in source routing an the probability that each path is chosen are base on the placement algorithms output. As shown in Figure 15, the ITAPs, etermine using the smooth egraation moel, satisfy the user emans to a great extent: aroun 8% houses have their emans completely satisfie when houses are ranomly place in 1*1 m 2, an all houses receive their emans when houses are ranomly place in 15 * 15 m 2. The better performance in the latter scenario comes from the fact that the larger separation among houses lowers interference among cross traffic. Note that even for the former case, we can further improve the clients throughput by over-provisioning. As shown in the same figure, with over-provisioning (assuming that each user s eman is 5 Kbps when the actual eman is 28 Kbps), most of the clients emans are satisfie. Probability area=15*15 area=1*1 area=1*1 (over-provision) Throughput (Kbps) Fig. 15. Valiation of general link moels: CDF of clients throughput, where N =5, WC =5Mbps, an w h =28Kbps 8h 2 H. Since ieal link an boune hop-count moels are more optimistic about the impact of interference, they are more suitable for the environments with efficient spectral use (e.g., when irectional antennas an/or multiple raios are use). As part of our future work, we plan to evaluate how well these two moels capture the impact of wireless interference uner such environments. VIII. FAULT TOLERANCE CONSIDERATION A practical solution to the ITAP placement problem shoul ensure Internet connectivity to all the houses in the neighborhoo, even in the presence of a few ITAP an house failures. In this section we present an enhancement to our problem by incorporating this fault tolerance constraint. Fault tolerance is achieve by proviing multiple inepenent paths from a house to ITAPs 1, an over-provisioning the elivery paths. Overprovisioning is a scheme that allocates more flow to a house than is esire, an therefore helps in proviing QoS guarantees even when there are a few failures. A. Problem Formulation Here we formulate the placement problem with the fault tolerance constraint. Let each house has one unit of eman, an inepenent paths to reach the ITAPs; the average failure probability of a path be p; an the over-provisioning factor be f (i.e., each inepenent path allocates f capacity to a house, an the total capacity allocate to a house by inepenent paths is f). Since for every house, there are inepenent paths from this house to ITAPs an the probability of failure of each path i is p, the probability that exactl of these paths fail is pi (1 1 These can be ifferent ITAPs since the ultimate goal is to provie Internet connectivitrrespective of which ITAP is use.