Optimizing the Placement of Integration Points in Multi-hop ..., Exercises of Wireless Networking

Download Optimizing the Placement of Integration Points in Multi-hop ... and more Exercises Wireless Networking in PDF only on Docsity! Optimizing the Placement of Integration Points in Multi-hop Wireless Networks Ranveer Chandra†, Lili Qiu
, Kamal Jain
, Mohammad Mahdian
Cornell University† Microsoft Research
Abstract Efficient integration of a multi-hop wireless network with the Internet is an important research problem. In a wire- less neighborhood network, a few Internet Transit Access Points (ITAPs), serving as gateways to the Internet, are de- ployed across the neighborhood; houses are equipped with low-cost antennas, and form a multi-hop wireless network among themselves to cooperatively route traffic to the Inter- net through the ITAPs. For both these applications, place- ment of integration points between the wireless and wired network is a critical determinant of system performance and resource usage. In this paper, we explore the place- ment problem under three wireless link models. For each link model, we develop algorithms to make informed place- ment decisions based on neighborhood layouts, user de- mands, and wireless link characteristics. We also extend our algorithms to provide fault tolerance and handle significant workload variation. We evaluate our placement algorithms and show that our algorithms yield close to optimal solu- tions over a wide range of scenarios we have considered. 1. Introduction Integrating multi-hop wireless networks to the Internet is an important research problem in wireless neighborhood networks. A few Internet Transit Access Points (ITAPs) are placed, relaying data from the wireless multi-hop network to the Internet and vice versa. This application requires ef- ficient bandwidth utilization at end nodes, which can be achieved through a careful placement of ITAPs. This pa- per explores efficient integration of multi-hop wireless net- works with the Internet by placing ITAPs at strategic loca- tions. Neighborhood networks are characterized by two impor- tant design constraints. They should be easy and cheap to deploy. Moreover, in order to be competitive to DSL or ca- ble providers, they should provide Quality of Service (QoS) guarantees to end users. To achieve both these constraints it is imperative to have an intelligent placement of ITAPs in the network. Any ITAP placement algorithm will have to (i) efficiently use wireless capacity, (ii) take into account the impact of wireless interference on network throughput, and (iii) be robust in face of failures and changes in user de- mands. There has been little previous work on this subject. In this paper, we investigate schemes to efficiently place ITAPs in a multihop wireless network. Our key contribu- tions are: • We formulate the ITAP placement problem under three wireless models. For each model, we develop algo- rithms to efficiently place ITAPs in the network. Our algorithms aim to minimize the number of required ITAPs while guaranteeing users’ bandwidth require- ments. We demonstrate the efficiency of the algorithms through simulation and analysis. • To enhance robustness, we present a fault tolerance version of the placement algorithm that provides band- width guarantees in the presence of failures. • We extend the algorithms to take into account vari- able traffic demands by developing an approximation algorithm to simultaneously optimize ITAP placement based on demands over multiple periods. This algo- rithm is very useful in practice since user demands of- ten exhibit periodic changes (e.g., diurnal patterns). 2. Related Work There has been a recent surge of interest in building wireless neighborhood networks [1] and [8]. [1] presents a scheme to build neighborhood networks using standard 802.11b Wi-Fi technology [22] by carefully positioning ac- cess points in the community. However, it requires a large number of access points. Moreover, it requires direct com- munication between machines and access points, which is difficult to meet in real terrains. Nokia’s Rooftop technol- ogy, presented in [8], provides broadband access to house- holds using a multi-hop approach that overcomes the short- comings of [1]. The idea is to use a mesh network model with each house deploying a radio, as considered in this paper. This radio serves the dual purpose of connecting to the Internet and routing packets for neighboring houses [4]. The deployment and management cost of Internet TAPs in such networks is significant, and so it is crucial to mini- mize the required number of ITAPs to provide QoS and fault tolerance guarantees. However, these problems are not ad- dressed in [1, 8]. There have been a number of interesting studies on plac- ing servers at strategic locations for better performance and efficient resource utilization in the Internet. For example, the authors in [21, 18, 24] examine placement of Web prox- ies or server replicas to optimize clients’ performance; and Jamin et al. [17] examines the placement problem for In- ternet instrumentation. Facility location problems are also related to ITAP placement problem, and have been con- sidered extensively in the fields of operation research (e.g., [20, 28]). Approximation algorithms with good worst case behavior have been proposed for different variants of this problem. The previous work on server placement or facility location cannot be applied to our context because they op- timize locality in absence of link capacity constraints. This may be fine for the Internet, but is not sufficient for wire- less networks since wireless links are often the bottlenecks. Moreover, the impact of wireless interference, and consid- erations of fault tolerance and workload variation make the ITAP placement problem very different from those studied earlier. The work closest to ours is the pioneering work in [3]. It aims to minimize the number of ITAPs for multi-hop neigh- borhood networks based on the assumption that ITAPs use a Time Division Multiple Access (TDMA) scheme to provide Internet access to users. However, TDMA is difficult to im- plement in multi-hop networks due to synchronization and channel constraints [2]. Furthermore, the proposed slotted approach might not utilize all the available bandwidth due to unused slots. In comparison, in this paper we look at more general and efficient MAC schemes, such as IEEE 802.11. Removing the TDMA MAC assumption yields completely different designs, and increases applicability of the result- ing algorithms. In summary, placing ITAPs under the impacts of link capacity constraints, wireless interference, fault tolerance, and variable traffic demands is a unique challenge that we aim to address in this paper. 3. 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 nodes on a plane, which we call houses. A house h is said to be successfully served, if its demand, wh, is sat- isfied by the ITAP placement. A house h is served by an ITAP i through a path between h and i. This path is al- lowed to pass through other houses, but any two consecu- tive points on this path must have wireless connectivity be- tween them. We are usually interested in the fractional ver- sion of this problem. That is, we consider the flexibility that a house is allowed to route its traffic over multiple paths to reach an ITAP. This problem can be modeled using the following graph- theoretic approach. Let H denote the set of houses and I denote the set of possible ITAP positions. We construct a graph G on the set of vertices H ∪ I by connecting two nodes if and only if there is wireless connectivity between them. The goal is to open the smallest number of ITAPs (de- noted by the set I ′), such that in the graph G[H ∪ I ′], one can route wh units of traffic from house h to points in I ′ si- multaneously, without violating capacity constraints on ver- tices and edges of the graph, where wh is the demand from house h. The edge capacity, Cape, in the graph denotes the capac- ity of a wireless link. In addition, each node also has an up- per bound on how fast traffic can go through it. Therefore, we also assign each node with a capacity, Caph. Usually Caph = Cape, as both represent the capacity of a wire- less link. (Our schemes work even when Caph = Cape, e.g., when a node’s processing speed becomes the bottle- neck.) Moreover, each ITAP also has a capacity limit, based on its connection to the Internet and its processing speed. We call this capacity, the ITAP capacity, Capi. In addition to edge and vertex capacities and house de- mands, another input to the placement algorithms is a wire- less connectivity graph (among houses). We can determine whether two houses have wireless connectivity using real measurements, and give the connectivity graph to our place- ment algorithms for deciding ITAP locations. In our perfor- mance evaluation, since we do not have wireless connec- tivity graphs based on real measurements, we instead de- rive connectivity graphs based on the protocol model [13]. In this model, two nodes i and j can communicate directly with each other if and only if their Euclidean distance is within a communication radius, CR. Given the position of all the nodes, we can easily construct a connectivity graph by connecting two nodes with an edge if their distance is within CR. However our placement algorithms can also work with other wireless connectivity models (e.g., phys- ical model [13] or based on real measurements). 3.1. Incorporating Wireless Interference There are several ways to model wireless interference. One approach is to use a fine-grained interference model based on the notion of a conflict graph, introduced in [16]. The main challenge of using the fine-grained interference model is high complexity (sometimes prohibitive), since for even a moderate-sized network the number of interference from houses to the ITAPs in S, and let fi denote the amount of traffic routed to ITAP i in this solution, where i ∈ S. As- sume that at a later time, a multiset S ∪ T of ITAPs are opened. Then, there is an optimal way of routing the maxi- mum total demands from houses to these ITAPs in which fi units of traffic is routed to ITAP i for every i ∈ S. Refer to [23] for the proofs of the above theorem and lemma. Based on Theorem 2, we have the following corol- lary. Corollary 4 Let N be the number of houses. The approx- imation factor of the greedy algorithm in the ideal link model is ln(N) when the capacities of edges and vertices are integer-valued and every house has either zero or one unit of demand. 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 model when every house has either zero or one unit of demand. Furthermore, even though in our model we allow fractional routing of the flow, our greedy algorithm always finds an in- tegral solution in this case, i.e., the demand from each house will be served 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 bound for heterogeneous demands. To make the worst-case bound tighter, we can normalize house demands, edge capacities, and node capacities before we apply the greedy placement algorithm. This yields a lower approximation factor, since D is reduced after normalization. Moreover, as we will show later in this section, in practice the greedy algorithm performs quite close to the optimal, and much better than the worst-case bounds, ln(D) or ln(N). 4.2. General Link Model The problem of efficient ITAP placement is more chal- lenging when the throughput along a path varies with the path length. This corresponds to the general link model in- troduced in Section 3.1. 4.2.1. Problem Formulation: We formulate the place- ment problem for the general link model as an integer lin- ear program shown in Figure 3. In this program x e,h,l,j de- notes the total amount of flow routed from house h to the ITAPs using a path of length l when edge e is the j’th edge along the path. Variable yi is an indicator of the number of ITAPs opened in the equivalence class i, and each house h has wh units of traffic to send. The throughput degradation function for a path of length l is denoted by g(l). L is an up- per bound on the number of hops on a communication path, and if there is no such upper bound, we set L = |H|. The other variables in the program are similar to the ones used by the program presented in Figure 1. minimize ∑ i∈I yi subject to ∑ e=(v,h′) xe,h,l,j = ∑ e=(h′,v) xe,h,l,j+1 ∀h, h′ ∈ H, h′ = h, l, j ∈ {1, . . . , L}, j < l ∑ e=(h,v),l xe,h,l,1 ≥ wh ∀h ∈ H ∑ h,l,j≤l g(l) xe,h,l,j ≤ Cape ∀e ∈ E(G) ∑ h′,e=(v,h),l,j≤l g(l) xe,h′,l,j ≤ Caph ∀h ∈ H ∑ h′,e=(v,i),l,j≤l g(l) xe,h′,l,j ≤ Capiyi ∀i ∈ I ∑ e=(u,i),l,j≤l xe,h,l,j ≤ whyi ∀i ∈ I, h ∈ H xe,h,l,j ≥ 0 ∀e ∈ E(G), h ∈ H, l, j ∈ {1, . . . , L}, j ≤ l yi ∈ {0, 1, 2, ...} ∀i ∈ I Figure 3: LP formulation for the general link model, where g(l) models throughput degradation with increasing hop- count. The following theorem is an immediate consequence of Theorem 1, as the ideal link model is a special case of the general link model, when g(l) = 1. Theorem 5 It is NP-hard to find a minimum number of ITAPs to cover a neighborhood for a general link model. 4.2.2. Our Approach of Greedy Placement: The high- level idea of the greedy algorithm is similar to the one pre- sented for the ideal link model. We iteratively select ITAPs to maximize the total user demands satisfied. The new chal- lenge is to determine a greedy move in this model. Unlike in the ideal link model, we cannot compute the total sat- isfied demands by modeling it as a network flow problem since the amount of flow now depends on the path length. As we will describe below, this computation can be done by solving a linear program, or by using a heuristic. Expensive algorithm for the general link model: Without making assumptions about g(l), we can com- pute the total satisfied user demands, for a given set I ′ of ITAPs, by solving a slightly modified LP prob- lem than the one in Figure 3. In this linear program, we re- place the variable yi by the number of occurrences of i in I ′ (This amounts to removing all the variables correspond- ing to edges ending in ITAP positions outside I ′ and remov- ing inequalities containing these variables). The objective will be to maximize ∑ h ∑ e=(h,v),l xe,h,l,1, which corre- sponds to maximizing the satisfied demands. We also mod- ify the second constraint to be ∑ e=(h,v),l xe,h,l,1 ≤ wh in order 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 [10], can only handle small-sized networks under this model due to the fast increase in the number of variables and con- straints with the network size. Below we develop more efficient algorithms for two spe- cial forms of g(l): (i) bounded hop-count: g(l) = 1 for all l ≤ k, and g(l) = ∞ for l > k, and (ii) smooth degrada- tion: gl = l for all l. Efficient algorithm for the bounded hop-count model: We can use the following greedy algorithm to find the total demands satisfied by a given set of ITAPs. The hop-count constraint suggests we should fa- vor short paths in the graph. Therefore, in each iter- ation, the algorithm finds the shortest path from de- mand points to opened ITAPs in the residual graph, routes one unit of flow along this path, and decreases the ca- pacities of the edges on the path by one in the resid- ual graph. This is continued until the shortest path found has length more than the hop-count bound. This algo- rithm is similar to the algorithm proposed in [14] for a sim- ilar problem. While this heuristic does not guarantee com- puting the maximum flow (so each greedy step is not local optimal), it works very well in practice as shown in Sec- tion 6.2.1. Efficient algorithm for the smooth throughput degra- dation model: When g(l) = l or throughputl = 1 l , the to- tal demands satisfied by a set of ITAPs are given by the ex- pression: maximize ∑ pi∈P 1 |pi| where P is a collection of edge-disjoint paths in the graph, and |p i| denotes the length of the path pi. Therefore to maximize this objective func- tion, our heuristic should prefer imbalance in path lengths, and this motivates the following algorithm. As the heuristic for the bounded hop-count model, in the smooth throughput degradation model we compute the to- tal satisfied demands by the selected ITAPs through itera- tively removing shortest paths in the residual graph. How- ever, we make the following modifications. First, since we no longer have bounds on hop-count, we continue pick- ing paths until there is no path between any demand point and any open ITAP. Second, to ensure the throughput fol- lows throughput(l) = 1/l, we compute the demand sat- isfied along each path p, denoted as SDp, according to the throughput function after we obtain all the paths. The to- tal satisfied demands are the sum of SDp over all paths p. Although this algorithm does not always find the maximum flow (so each greedy step is not local optimal), it yields very good performance as shown in Section 6.2.2. 4.3. Alternative Algorithms In the rest of this paper, we compare our greedy place- ment algorithm to four alternative approaches. 4.3.1. Augmenting Placement: The idea of the augment- ing placement algorithm is similar to the greedy algorithm. The main difference is that in the augmenting algorithm we do not make a greedy move; instead we are satisfied with any ITAP that increases the total amount of demand sat- isfied. More specifically, we search over the set of possi- ble ITAP locations, and open the first ITAP we see that re- sults in an increase in the amount of satisfied demand when opened together with the already opened ITAPs. The augmenting placement algorithm can be applied to all three wireless link models with the following difference. In the ideal link model, we compute the total amount of de- mand satisfied under a given set of ITAPs by finding the maximum flow in the graph; whereas in the general link models, we use the heuristics described in Section 4.2.2 to derive the total amount of demand satisfied. 4.3.2. Clustering-based Placement: We compare our placement algorithms to the clustering-based scheme, pro- posed in [3]. The basic idea of the algorithm is to partition the network nodes into a minimum number of disjoint clus- ters, and place an ITAP in each cluster. We use the Greedy Dominating Independent Set (DIS) [3] heuris- tic to determine a set of clusterheads, which are used as possible ITAP locations. The nodes are then clus- tered to ensure that each node is associated with the closest clusterhead, and a shortest path tree rooted at the clus- terhead is used for sending packets from and deliver- ing packets to the cluster. The cluster is further divided into sub-clusters if either the weight or relay-load con- straints are violated. The weight constraint specifies that an ITAP can serve nodes as long as the sum of their de- mands does not exceed the capacity of the ITAP, and the relay-load constraint specifies an upper bound on the maxi- mum flow that can go through a node in the neighborhood cluster. We refer the reader to [3] for more details of this al- gorithm. To apply the clustering-based algorithm for the ideal link model, in our simulations we use the ITAP capacity instead 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 ca- pacity in our simulations. This ensures a fair comparison of the clustering algorithm with our placement schemes. To apply the algorithm to the bounded-hop count model, we make the following modification. We divide a cluster into sub-clusters not only when the weight or relay-load constraints are violated, but also when the distance between any node and its clusterhead exceeds the hop-count thresh- old. The algorithm, however, does not apply to the smooth throughput degradation model. 4.3.3. Random Placement: This algorithm randomly places an ITAP at a house iteratively until all the user de- mands are satisfied. To avoid wasting resource, it ensures that each house has at most one ITAP. This approxi- mates un-coordinated deployment of ITAPs in a neighbor- hood, and gives a baseline to evaluate the benefits of the more sophisticated algorithms presented above. As the augmenting algorithm, there are three variants of random placement algorithms for different wireless link models. They differ in how we compute the total demand satisfied under a given set of ITAPs. We run the maximum flow algorithm to compute the satisfied demand under the ideal link model, and apply the heuristics in Section 4.2.2 to compute the satisfied demand under the general link mod- els. 4.3.4. Lower Bound: It is useful to compare our algo- rithms with the optimal solution. However, our problem is NP-hard, and it is too expensive to derive an optimal solu- tion. Therefore we compare our algorithms with the lower bounds. We derive the lower bound by relaxing the inte- ger constraints on yi in the LP (in Figure 1) and solving the relaxed LP problem using cplex [10]. The lower bound is a useful data point to compare with, as it gives an up- per bound on the difference between a practical algorithm and the optimal. We use the same scheme to derive lower bounds for all three link models. Note that ideally we would like to derive the lower bound for the general link models by relaxing the integrality constraint in Figure 3, and solving the relaxed linear program. However, although in theory linear pro- grams can be solved in polynomial time, we were unable to solve the program in Figure 3 for large networks due to the memory constraints. So in our performance evaluation sec- tion, we use the solution to the LP formulation of Figure 1 for the ideal link model, as the lower bound for the general case too. This lower bound is always correct, since the ideal link model is a relaxation of the general model. However, it might not be tight, since it ignores the throughput degra- dation with hop count, and therefore requires fewer ITAPs than necessary. However, in Section 6.2.1 and Section 6.2.2 we show that the results from our greedy and augmenting algorithms are still close to these loose lower bounds. 5. Validation To validate the wireless link models used in this paper, we run simulations in Qualnet [25], a commercial network simulator. More specifically, given a neighborhood layout, the placement algorithms determine the ITAP locations and the set of paths each house uses to reach the ITAPs. We use the same neighborhood layout and ITAP locations in the simulations. Every node in the simulation uses an omni- directional antenna and 802.11b MAC, with the communi- cation range and interference range being 195 meters and 376 meters, respectively. Every house sends CBR traffic to the ITAPs at the rate specified by the placement algo- rithms’ output. To support multi-path routing, we imple- mented probabilistic source-routing in Qualnet, where the 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 60 80 100 120 140 160 180 200 220 Pr ob ab ili ty Throughput (Kbps) area=1500*1500 area=1000*1000 area=1000*1000 (over-provision) Figure 4: Validation of general link models: CDF of clients’ throughput, where N = 50, WC = 5Mbps, and wh = 208Kbps ∀h ∈ H . paths used in source routing and the probability that each path is chosen are based on the placement algorithms’ out- put. As shown in Figure 4, the ITAPs, determined using the smooth degradation model, satisfy the user demands to a great extent: around 80% houses have their demands completely satisfied when houses are randomly placed in 1000*1000 m2, and all houses receive their demands when houses are randomly placed in 1500 * 1500 m 2. The bet- ter performance in the latter scenario comes from the fact that the larger separation among houses lowers interfer- ence 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 demand is 500 Kbps when the actual demand is 208 Kbps), most of the clients’ demands are satisfied. Since ideal link and bounded hop-count models are more optimistic about the impact of interference, they are more suitable for the environments with efficient spectral use (e.g., when directional antennas and/or multiple radios are used). As part of our future work, we plan to evaluate how well these two models capture the impact of wireless inter- ference under such environments. 6. Performance Evaluation In this section, we evaluate the performance of differ- ent placement algorithms using various network topologies, house demands, and link models. 6.1. Performance Under the Ideal Link Model First, we look at the performance under the ideal link model under various scenarios. We use the following nota- tions in our discussion. • N : the number of houses • WC: a wireless link’s capacity • IC: an ITAP’s capacity • CR: communication radius • HR: average inter-house distance • wh: house h’s demand hop-count threshold=3 0 20 40 60 80 100 120 0 50 100 150 200 250 communication radius (meters) # IT A P s greedy augment clustering random lower bound Figure 9: Bounded hop-count model: a real neighborhood topology for various communication radii, where N = 105, WC = 6, IC = 100, hop-count threshold = 3, and the house demands follow a Zipf distribution. 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 communication radius # IT A P s greedy augment random lower bound Figure 10: Smooth throughput degradation model: varying the communication radius, where N = 100, WC = 6, IC = 100, and wh = 1 ∀h ∈ H . tion radius is between 5 and 20 (The average inter-house is around 5.). This can be explained as follows. When the radius is very small, most houses are disconnected from one another. So, the number of ITAPs required is nearly the number of houses. When the radius is very large, most houses are reachable from one another within one or few hops, and the number of ITAPs required is close to 1. How- ever, for the practical scenario with medium communica- tion radius, the gap between different algorithms is most significant, especially between the random placement and the other two. Note that the lower bound, which is derived by ignoring the impact of hop-count on throughput, is more loose for this scenario. Even then the greedy is still compet- itive when compared with these loose lower bounds. Real neighborhood topology: Figure 11 shows the re- sults from the real neighborhood topology. As we can see, the greedy placement continues to perform well, yielding close to optimal performance. 7. Practical Considerations We now present algorithms for handling two practical requirements: providing fault tolerance and handling work- load variation. 0 20 40 60 80 100 120 0 50 100 150 200 250 communication radius (meters) # IT A P s greedy augment random lower bound Figure 11: Smooth throughput degradation model: a real neighborhood topology for various communication radii, where N = 105, WC = 6, IC = 100, and the house de- mands follow a Zipf distribution. 7.1. Providing Fault Tolerance A practical solution to the ITAP placement problem should ensure Internet connectivity to all the houses in the neighborhood, even in the presence of a few ITAP and house failures. Here we present an enhancement to our algorithm by incorporating this fault tolerance constraint. Fault tol- erance is achieved by providing multiple node independent paths from a house to ITAPs1, and over-provisioning the de- livery paths. Over-provisioning is a scheme that allocates more flow to a house than its demand, and therefore helps in providing QoS guarantees even when there are a few fail- ures. 7.1.1. Problem Formulation: Let each house have one unit of demand, and d independent paths to reach the ITAPs; the average failure probability of a path be p; and the over- provisioning factor be f (i.e., each independent path allo- cates f d capacity to a house, and the total capacity allocated to a house by d independent paths is f ). Since for every house, there are d independent paths from this house to ITAPs and the probability of failure of each path is p, the probability that exactly i of these paths fail is ( d i ) pi(1 − p)d−i. In this case, the amount of traffic that can be delivered is min( (d−i)f d , 1). Therefore, the ex- pected fraction of the traffic from a house that can reach an ITAP, S(f, p, d), is given by the following formula. S(f, p, d) = d∑ i=0 ( d i ) pi(1 − p)d−i min( (d − i)f d , 1). Given the expected guarantee desired by the home users, S(f, p, d), we can use the above expression to derive the overprovision factor, f , based on path failure probability and the number of independent paths. We now provide fault tolerant LP formulations for the ideal and general link mod- els. 1 These can be different ITAPs since the ultimate goal is to provide In- ternet connectivity irrespective of which ITAP is used. minimize ∑ i∈I yi subject to ∑ e=(v,h′) xe,h = ∑ e=(h′,v) xe,h ≤ wh ∀h, h′ ∈ H, h′ = h ∑ e=(h,v) xe,h − ∑ e=(v,h) xe,h ≥ whd ∀h ∈ H f d ∑ h xe,h ≤ Cape ∀e ∈ E(G) f d ∑ h′,e=(v,h) xe,h′ ≤ Caph ∀h ∈ H f d ∑ h′,e=(v,i) xe,h′ ≤ Capiyi ∀i ∈ I ∑ e=(v,i) xe,h ≤ whyi ∀i ∈ I, h ∈ H xe,h ≥ 0 ∀e ∈ E(G), h ∈ H yi ∈ {0, 1, 2, ...} ∀i ∈ I Figure 12: LP formulation for the ideal link model with fault tolerance constraints, where d is the number of independent paths, and f is the over-provision factor. Ideal Link Model with the Fault Tolerance Con- straint : Figure 12 provides an LP formulation of the fault tolerant problem for the ideal case, i.e. when through- put is independent of the path length. For each edge e and each house h, the variable xe,h indicates the amount of flow from h to ITAPs that is routed through e. Also, for each ITAP i, the variable yi denotes the number of ITAPs opened in equivalence class i. The above integer LP is sim- ilar to the one in Section 4.1.1. The differences are as follows: (i) the constraint ≤ wh added to the first inequal- ity, (ii) a change in the second constraint from wh to whd in the amount of flow originating from each house, and (iii) a multiplicative factor of f d on the left-hand side of the ca- pacity constraints (since the amount of capacity each path allocates to each house is f d ). The first modification en- sures that the flow from each house is served by inde- pendent paths; (ii) and (iii) are for the over-provisioning purpose. Similarly we can formulate integer LP for general link models with the fault tolerance constraint. Refer to [23] for details. For all link models, we have Theorem 6. Theorem 6 It is NP-hard to find a minimum number of ITAPs required to cover a neighborhood while providing fault tolerance. 7.1.2. Placement Algorithms: The greedy, augmenting and random placement algorithms are based on the same idea described in the previous sections of this paper. How- ever, they differ in the way they compute the total demands supported by a given set of ITAPs. For the ideal case, we compute the satisfied demands by slightly modifying the LP in Figure 12, and solving the resulting LP. The objective function is changed to be maximizing ∑ h( ∑ e=(h,v) xe,h−∑ e=(v,h) xe,h), which corresponds to maximizing the sup- ported demands. The variables yi are replaced by the num- ber of occurrences of i in I ′. Furthermore, the second con- straint is changed to ∑ e=(h,v) xe,h− ∑ e=(v,h) xe,h ≤ whd in order to limit the maximum flow from a node. For the general link model, we compute the satisfied demands by applying similar modifications. We compared the above algorithms to the lower bound, derived by relaxing the integrality constraint and solving the relaxed linear program. Our evaluation [23] shows that for all algorithms the number of ITAPs required increases lin- early with the number of independent paths. Moreover, the results of the greedy algorithm are very close to the lower bound, and significantly better than the other two. 7.2. Handle Workload Variation In practice, user demands change over time, and often exhibit diurnal patterns [6, 19, 26]. Since it is not easy to change ITAP locations once they are deployed, a good ITAP placement should handle demands over all periods. In this section, we describe and evaluate two approaches to han- dle variable workloads. While our discussion focuses on the non fault-tolerant version of the placement problems, the ideas carry over easily to the fault-tolerant version as well. One approach to take into account workload change is to provision ITAPs based on the peak workload. That is, if w[h][t] denotes the demand of house h at time t, we use maxt w[h][t] as the demand for house h, and feed this as an input to the placement algorithms described in the previous sections. We call this approach peak load based placement. This algorithm is simple, but may sometimes be wasteful, e.g., when different houses’ demands peak at dif- ferent times. To improve efficiency without sacrificing user perfor- mance, we now explore how to optimize ITAP locations for demands over multiple time intervals. More formally, the problem can be stated as follows. Each house h has de- mand w[h][t] at time t. Our goal is to place a set of ITAPs such that at any time t, they can serve all the demands gen- erated at t, i.e., w[h][t] for all h’s. Here we describe a greedy heuristic with a logarithmic worst-case bound for the ideal link model. The same idea applies to other link models. The high-level idea is to it- eratively place the ITAP such that together with the al- ready opened ITAPs it maximizes the total demands served. Unlike in the previous section, here the total demands in- clude demands over multiple time intervals. More specifi- cally, we place an ITAP such that it maximizes ∑ t∈T SDt, where SDt is the total satisfied demands at time t. This can be computed by changing the greedy algorithm of Section 4.1.2 as follows. In every iteration, for every j ∈ I and t ∈ T , we construct the graph G′ as in the algorithm of Sec- tion 4.1.2 based on the demands at time period t. Then we compute the maximum flow fj,t in this graph. After these computations, we pick the ITAP j that maximizes ∑ t fj,t, and open it. We call this algorithm multiple-demand-based greedy placement (M-greedy, for short). In the following theorem, we prove a worst-case bound on the M-greedy’s performance in the ideal link model. Theorem 7 Consider the ITAP placement problem in the ideal link model with integral demands and capacities, and let Dt be the total demand in period t. The approxima- tion factor of the M-greedy algorithm for this problem is at most ln( ∑ t Dt). In other words, if the optimal algorithm requires K ITAPs to serve demands over L time periods, then the M-greedy requires at most K ln( ∑ t Dt) ITAPs. Refer to [23] for the proof. Based on the above theorem, we have the following corollary. Corollary 8 Let L denote the total number of periods, and N denote the number of houses. The approximation factor of the M-greedy in the ideal link model is ln(LN), when the capacities of edges and vertices are integer-valued and ev- ery house has either zero or one unit of demand at any time t. This is easy to see because ∑ t Dt ≤ LN . The approximation factor of the greedy placement using the peak load is at most a factor of ∑ t(ln Dt) (refer to [23] for details). When Dt = Dm for all t’s, its cost is at most L ln(Dm). This is roughly L times the approximation fac- tor of the M-greedy algorithm proved above. In addition to the worst-case analysis, we evaluate the ef- fectiveness of the algorithms empirically, and observe that the number of ITAPs required to serve demands at two dif- ferent periods using M-greedy is only slightly higher than the maximum number of ITAPs required to serve either of the two periods. In the interest of space, we refer readers to [23] for details. 8. Conclusions In this paper we look at the problem of efficient ITAP placement to provide Internet connectivity in multi-hop wireless networks. We make three major contributions in this paper. First, we formulate the ITAP placement prob- lem under various wireless models, and design algorithms for each model. Second, we address several practical is- sues when using these algorithms. In particular, we extend the placement algorithms to provide fault tolerance and to handle variable user demands. These two enhancements im- prove robustness of our placement schemes in face of fail- ures and demand changes. Third, we demonstrate the effi- ciency of our placement algorithms using analysis and sim- ulations, and show that the greedy algorithms give close to optimal solutions over a variety of scenarios we have con- sidered. To our knowledge this is the first paper that looks at the ITAP placement problem for general MAC schemes. References [1] 802.11b community network list. http://www.toaster.net/wireless/ community.html. [2] S. Basagni, M. Conti, S. Giordano, and I. Stojmenovi, editors. Mo- bile Ad Hoc Networking. Wiley-IEEE Press, Jul. 2004. [3] Y. Bejerano. Efficient integration of multi-hop wireless and wired networks with QoS constraints. In ACM MOBICOM, Sep. 2002. [4] D. A. Beyer, M. Vestrich, and J. J. Garcia-Luna-Aceyes. The rooftop community network: Free, High-Speed network access for commu- nities. In The First 100 Feet: Options for Internet and Broadband Access. MIT Press: Cambridge, 1999. [5] L. Breslau, P. Cao, L. Fan, G. Phillips, and S. Shenker. Web caching and Zipf-link distributions: Evidence and implications. In Proc. of IEEE INFOCOM 99, Mar. 1999. [6] M. Chesire, A. Wolman, G. M. Voelker, and H. M. Levy. Mea- surement and analysis of a streaming-media workload. In Proc. of 3rd USENIX Symposium on Internet Technologies and Systems, Mar. 2001. [7] R. R. Choudhury, X. Yang, R. Ramanathan, and N. H. Vaidya. Using directional antennas for medium access control in ad hoc networks. In Proc. of ACM MOBICOM, Sept. 2002. [8] Wireless networking reference - community wiress/rooftop systems. http://www.practically- networked.com/tools/wireless articles community.htm. [9] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduc- tion to Algorithms (Second Edition). MIT Press, Cambridge, 2001. [10] CPLEX version 7.5.0. http://www.ilog.com/products/cplex/. [11] T. ElBatt and B. Ryu. On the channel reservation schemes for ad- hoc networks: Utilizing directional antennas. In 5th IEEE Int’l Sym- posium on Wireless Personal Multimedia Communications, 2002. http://www.wins.hrl.com/projects/adhoc/d/elbatt wpmc02.pdf. [12] M. Gerla, R. Bagrodia, L. Zhang, K. Tang, and L. Wang. TCP over wireless multi-hop protocols: Simulation and experiments. In Proc. of 1999 IEE International Conference on Communications (ICC), Jun. 1999. [13] P. Gupta and P. R. Kumar. The capacity of wireless networks. IEEE Transactions on Information Theory, 46(2), Mar. 2000. [14] V. Guruswami, S. Khanna, R. Rajaraman, B. Shepherd, and M. Yan- nakakis. Near-optimal hardness results and approximation algo- rithms for edge-disjoint paths and related problems. In Proc. of the 31st ACM Symposium on Theory of Computing (STOC), pages 19– 28, 1999. [15] G. Holland and N. Vaidya. Analysis of TCP performance over mo- bile ad hoc networks. In Proc. of ACM Mobicom 99, 1999. [16] K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qiu. Impact of inter- ference on multi-hop wireless network performance. In Proc. ACM MOBICOM, Sept. 2003. [17] S. Jamin, C. Jin, Y. Jin, D. Raz, Y. Shavitt, and L. Zhang. On the placement of Internet instrumentation. In IEEE INFOCOM, Mar. 2000. [18] S. Jamin, C. Jin, A. Kurc, D. Raz, and Y. Shavitt. Constrained mirror placement on the Internet. In Proc. of IEEE Infocom, 2001. [19] D. Kotz and K. Essien. Analysis of a campus-wide wireless network. In Proc. of ACM MOBICOM, Sept. 2002. [20] A. Kuehn and M. Hamburger. A heuristic program for locating ware- houses. Management Science, 9:643–666, 1963. [21] B. Li et al. On the optimal placement of Web proxies in the Internet. In Proc. of IEEE INFOCOM, 1999. [22] L. M. S. C. of the IEEE Computer Society. Wireless LAN medium access control (MAC) and physical layer (PHY) specifications. IEEE Standard 802.11, 1999. [23] L. Qiu, R. Chandra, K. Jain, and M. Mahdian. On the placement of integration points in multi-hop wireless net- works. In MSR Technical Report MSR-TR-2004-43, 2004. http://www.research.microsoft.com/˜liliq/papers/pub/msrtr- placement2004.ps. [24] L. Qiu, V. N. Padmanabhan, and G. Voelker. On the placement of Web server replicas. In Proc. of IEEE Infocom, 2001. [25] Qualnet. http://www.scalable-networks.com/. [26] S. Saroiu, K. P. Gummadi, R. J. Dunn, S. D. Gribble, and H. M. Levy. An analysis of Internet content delivery systems. In Proc. of the 5th Symposium on Operating Systems Design and Implementa- tion (OSDI’2002), Dec. 2002. [27] J. L. Sobrinho and A. S. Krishnakumar. Quality of service in ad hoc carier sense multiple access wireless networks. In JSAC, 1999. [28] V. V. Vazirani. Approximation Algorithms. Springer-Verlag, 2001.