Efficient Channel Allocation for Wireless Local-Area Networks

Transcription

1 1 Efficient Channel Allocation for Wireless Local-Area Networks Arunesh Mishra, Suman Banerjee, William Arbaugh Abstract We define techniques to improve the usage of wireless spectrum in the context of wireless local area networks (WLANs) by defining better channel allocation methods among interfering access points (APs). We do this by identifying new ways of channel re-use that are based on realistic interference scenarios in WLAN environments. This problem can typically be modeled as a graph coloring algorithm, where the APs represent vertices and the number of available channels represent colors. We show in this paper that such a formulation is unable to gain maximal benefits in WLANs. Instead we define a weighted variant of the graph coloring problem that takes into account realistic channel interference observed in wireless environments, as well as the impact of such interference on wireless users. We prove that the weighted graph coloring problem is NP-hard and propose scalable and fault tolerant distributed algorithms that achieve significantly better performance than existing techniques for channel allocation. Through detailed simulations we study the performance of the algorithms under various topologies and channel conditions. In particular, we show that the techniques achieve up to 45.5% reduction in the number of clients affected by interference for IEEE 82.11b/g networks with sparse topologies. The algorithms scale well with the size and density of the underlying topology. We also evaluate the algorithms on an in-building wireless testbed (consisting of 2 APs) and demonstrate that the algorithms achieve a 4% reduction in the interference by using partially overlapping channels when an insufficient number of nonoverlapping channels are available. I. INTRODUCTION Wireless local area networks (WLANs) have gained importance in the recent years as an Internet-access technology. As competition has driven down costs of WLAN equipment, wireless Internet access mechanisms are increasingly available in numerous public hot spots like coffee-shops, airports, and hotels. WLANs operate in portions of the frequency spectrum allotted by a regulatory body, e.g., the Federal Communications Commission in the USA, for unlicensed use. For example the 82.11b and 82.11g based WLANs operate in the 2.4 GHz frequency band while the 82.11a based WLANs operate in the 5 GHz frequency band. Such unregulated spectrum allocation has contributed to significant popularity and growth of WLANs. With the increasing deployment of such WLANs, network administrators are faced with an emerging challenge of distribution of the key shared resource (bandwidth) amongst the numerous users. In this paper we focus on a specific resource sharing problem in the context of based WLANs channel allocation. A. Mishra and W. Arbaugh are with the Department of Computer Science, University of Maryland, College Park, MD 2742, USA. s: {arunesh,waa}@cs.umd.edu. S. Banerjee is with the Department of Computer Sciences, University of Wisconsin, Madison, WI 5376, USA. suman@cs.wisc.edu Consider an in-building wireless environment in which multiple Access Points (APs) are operational. Each AP operates on an administrator-specified channel. In WLANs, the wireless card of a user scans the wireless medium to identify the access point with the strongest signal and associates with it. In order to reduce interference between different APs in the same physical neighborhood administrators conduct detailed Radio Frequency (RF) site surveys, often using spectrum analyzers, prior to setting up APs within the building and assigning specific channels to them [1]. Each WLAN standard defines a fixed number of channels for use by APs and mobile users. For example, the 82.11b standard defines a total of 14 frequency channels of which 1 through 11 are permitted in the US, 1 through 13 are permitted in Europe and only channel 14 is permitted in Japan. An important concept to note regarding these channels is that the channel actually represents the center frequency that the transceiver within the radio and AP uses (e.g., GHz for channel 1 and GHz for channel 2). There is only 5 MHz separation between the center frequencies, and an 82.11b signal occupies approximately 3 MHz of the frequency spectrum. The signal falls within about 15 MHz of each side of the center frequency. As a result, an 82.11b signal on any channel overlaps with several adjacent channel frequencies causing interference (also known as adjacent channel interference). This leaves only three channels (channels 1, 6, and 11) for the US that can be used simultaneously without causing interference. As a basic design rule, APs within range of each other are set to to different nonoverlapping channels. The current best practice for such nonoverlapping channel assignment is performed as follows: Each AP periodically checks other data transmissions in the channel it is using. If the volume of traffic in that channel (from other APs or clients of other APs) is greater than a threshold, then the first AP tries to move over to a less congested channel. We call this technique, Least Congested Channel Search (). In this paper we contend that with the continued growth of WLANs, a channel assignment scheme based solely on the rule will not be efficient. Given the unlicensed nature of WLAN technologies and decreasing costs of APs, the number of APs located in a physical neighborhood has proliferated. In many cases administrators increase the number of APs within a building to improve the wireless coverage. Additionally multiple organizations co-resident in the same building deploy independent wireless LANs, and the channel assignments made for the respective APs are made independent of each other. Hence the assignment of channels to this potentially large set of APs needs to be carefully coordinated, or else the broadcast nature of WLANs can lead to serious performance degradation of the

2 wireless users. Considerations for Efficient Channel Allocation We define scalable distributed algorithms for the channel assignment problem that tries to optimize user performance in wireless LAN environments with multiple APs. Our formulation and solution to the WLAN channel assignment problem addresses two important problems which, to the best of our knowledge, have not been addressed in prior literature. They are: 3 to a re-assignment of channels as the overlap regions become populated with users 1. Weighted channel assignments: A channel assignment problem is typically modeled as a graph-coloring problem there is a vertex on the graph corresponding to each AP, an edge on this graph represents potential interference, and the colors represent the number of non-overlapping channels. A goal of the channel assignment problem is to cover all APs (vertices) with the minimum number of channels (colors) such that no two adjacent APs (vertices) use the same channel (color). This is the minimum graph coloring problem. 2 1 AP1 Region X AP2 Region Y AP3 AP6 AP AP4 AP5 The un- Fig. 1. Channel assignment should be based on user performance. shaded circles indicate the interference radius of each AP. Dynamic channel re-use and Discovery of hidden-aps: Consider the scenario depicted in Figure 1. There are three APs, namely AP 1, AP 2, and AP 3. In this figure, mobile users 1 3 are associated with AP 1, mobile users 4 7 are associated with AP 2, and mobile users 8 12 are associated with AP 3. Transmissions from AP 1 and AP 2 do not interfere with each other. Hence we would ideally like to assign both of these APs to the same channel to maximize channel reuse. Note that a simple RF site survey (conducted both at AP 1 or AP 2 ) will also arrive at the same decision. Transmissions from AP 2 and AP 3 also do not interfere with each other. However, the clients of AP 2 (4 7) and AP 3 (9 12) are in close proximity of each other. Hence if AP 2 and AP 3 operate on the same wireless channel, only one of these APs can be actively involved in communication with a client in any given instant. This is the classic hidden terminal problem in wireless networks [2], [3]. The Distributed Coordinated Function (DCF) in 82.11b WLAN technology will handle such hidden terminals (hidden APs) by using the four-way handshake (RTS-CTS-Data-Ack), effectively reducing the data throughput achieved by the mobile users. An even more efficient solution is to assign these two APs to two different non-overlapping channels, if that is possible. Unfortunately, an RF site survey at either of the AP locations or the heuristic does not expose this information to the site administrator. The decision to assign two different APs to the same wireless channel significantly depends on the potential interference experienced by the mobile users in an overlap region, i.e., Regions X and Y in Figure 1. If both these regions are empty, then all three APs can be assigned to the same channel to maximize channel reuse. Our proposed solutions dynamically identifies opportunities for channel reuse when such overlap regions are devoid of mobile users, but quickly reverts back Fig. 2. All APs are in interference range of each other and assuming we have only three non-overlapping channels, we need to assign overlapping channels to APs in this case. However, the number of non-overlapping channels available in a specific WLAN technology is bounded. For example, in 82.11b in the US there are only three independent, nonoverlapping channels, 1, 6, and 11. Consider the scenario in Figure 2 in which the four APs (AP 4, AP 5, AP 6, and AP 7 ) are in close proximity of each other. There are 9, 4, 3, and 1 mobile clients associated with them respectively. In such scenario if we restrict our channel choices to only the non-overlapping channels (1, 6, and 11) two APs will be assigned the same channel and will experience significant interference. We demonstrate in this paper that better performance can be achieved by assigning partially overlapping channels to the APs. One possible assignment of channels to APs is: AP 4 to channel 1, AP 5 to channel 6, AP 6 to channel 11, and AP 7 to channel 9. (The channels are separated by gaps of 5,1, and 1 intervening channels.) Clearly clients of AP 5, AP 6, and AP 7, will experience some interference. AP 4 encounters no interference from other APs and their clients. This is desirable because the number of clients it is currently supporting is greater than the total number of clients in all the other APs. However, it is possible to consider alternative channel assignments, e.g. AP 4 to channel 1, AP 5 to channel 11, AP 6 to channel 5, and AP 7 to channel 8. (The channels are separated by gaps of 3,2, and 2 intervening channels.) In this scenario there is some interference between each pair of APs. However, the interference between AP 5, AP 6, and AP 7, is reduced. It is possible that such an assignment leads to better aggregate 1 We ignore small timescale, transient migrations of user populations.

3 performance for all clients. Based on these observations, we define a weighted variant of the graph-coloring problem, in which it is permissible to allocate overlapping channels to neighboring APs. Our goal in this variant is to minimize the impact of such overlapping channel assignments between neighboring APs on user performance. We discuss the exact formulation of this problem in Section II. Unlike the RF site survey based static techniques, our proposed solution enables self-configuration of channel assignment to APs based on continuously evolving channel conditions in the wireless environment. We present two techniques in this paper that are applicable to a wide range of WLANs technologies, e.g., 82.11b, 82.11a, 82.11g, etc. The first technique does not require any collaboration among the APs and can be applied to a wireless network formed by APs belonging to different WLANs. The technique assumes that the APs are greedy in nature they try to minimize the interference within their area of coverage 2. We also present a second technique that drastically reduces the total number of clients suffering interference in the network as a whole. However, this technique requires collaboration among the APs. This is particularly suited for efficient channel allocation for a single wireless network, or for multiple wireless networks if the administrators are willing to cooperate. Through simulations, we show that the two techniques presented in this paper dramatically improve interference. In particular, with 3 non-overlapping channels, the techniques achieve a 45.5% reduction in interference for sparse topologies. We also show that even if only 1% of the total APs use the techniques, the network as whole still achieves up to 4% reduction in the clients suffering interference. We also demonstrate similar performance results on an in-building wireless network. The rest of the paper is structured as follows: In Section II we present the formulation of our channel assignment problem for wireless LANs. In section III, we discuss a graph theoretic model for interference and provide distributed algorithms in section IV. Section V discusses the simulation results, and the experiments are discussed in section VI. In Section VII we describe the related work. We conclude in section VIII. II. IMPROVING CHANNEL RE-USE The objective of this work is to define an assignment of channels to WLAN APs in such a way that it maximizes data throughputs of wireless users. We first examine the current best practices for channel allocation in WLAN environments. A. Current Best Known Algorithms Current best practices for channel assignment recommend a manual placement of access points and channel selection [1], aided with radio-map tools. Access points, however, provide a method of autonomously searching for the least congested channel [4]. Below we discuss the two approaches and their shortcomings: 2 A rogue AP can always thwart any algorithmic technique for efficient channel assignment to all other neighboring APs in its vicinity. 1) Least Congested Channel Search (): In this technique, each AP monitors packet transmissions in the same channel. When it detects such transmissions from other APs or clients associated to other APs (within its vicinity), it tries to scan for an alternative lesscongested channel. In order to be able to search for the least congested channel (channel with least number of other APs assigned in the vicinity), the AP should be within the transmission range of stations associated to another access point. We will show in this section that using such an algorithm, an AP cannot detect many scenarios of conflict between neighboring APs. 2) Radio-map tools: Site administrators also use radio-map tools to assign channels to APs. This method is not autonomous, and hence requires a manual re-configuration of the network as the APs change position or the wireless characteristics of the environment change (e.g. on adding new furniture in an office). Also this assumes that the administrator has control over the spectrum, i.e, if other networks were to co-exist, this approach can become very cumbersome and inefficient. B. Our Opportunities for Efficient Channel Re-use The techniques proposed in this paper significantly improve the channel allocation mechanisms for WLANs by automatically detecting opportunities for greater channel re-use when possible, and also correctly handling situations that would lead to greater interference between neighboring APs and their clients. To precisely define our channel assignment approach, we first define the notion of interference between wireless transmitters and receivers that will be relevant in the rest of the paper. In particular we look at two interference effects: (i) co-channel interference, which will be useful in addressing dynamic channel re-use and discovery of hidden APs, and (ii) adjacent channel interference that will be important in defining and solving the weighted variant of the graph coloring problem. 1) Detecting and Avoiding Co-Channel Interference: Cochannel interference refers to interactions of multiple wireless transmissions on the same channel (with the same center frequency). We borrow some of the terminology used in [5]. When a signal is propagated from a transmitter to a receiver, the signal is correctly received at the receiver depending on the received power at the receiver. Given a transmission power, the receiving power is mostly decided by path loss characteristics over the transmitter-receiver distance. For the simplicity of discussion, we will assume an openspace environment, in which the path loss of a signal is usually modeled as a two-way ground model. Let d is the distance between the receiver and the transmitter. When the transmitter is close to the receiver (e.g. within the Freznel zone [6]), the received signal power is inverse proportional to d 2. When their distance is larger (e.g. outside of Freznel zone), the receiving signal power is then inverse proportional to d 4 [6]. (Note that our algorithms are independent of the specific signal attenuation model in the physical environment.) According to this model, in the open space environment, the received power P r of a signal from a sender at distance d is

4 2R 2R 2R 2R R BSS AP 1 R C1 C2 AP 2 R BSS AP 1 R C1 C2 AP 2 R BSS AP 1 R AP 2 C1 C2 R BSS AP 1 R C1 C2 AP 2 D > R BSS D [R to R + R] BSS BSS D [ 2R to R BSS ] D < 2R (a) No Conflict (b) Type-1 Conflict (c) Type-2 Conflict (d) Type-3 Conflict Fig. 3. Figure shows four different situations with APs AP 1 and AP 2 sharing the same channel. The innermost circle (of radius R) around an AP indicates its transmission radius. The next larger circle (of radius 2R) indicates the maximum transmission radius among all mobile users associated with this AP. The outermost circle indicates the maximum interference radius of users associated with the AP. given by h 2 t P r = P t G t G h2 r r d k (1) In equation 1, G t and G r are the antenna gains of the transmitter and the receiver respectively. h t and h r are the height of both antennas. P t is the transmission power at the sender. The path loss, parameter k, typically is a value between 2 and 4. A signal arriving at a receiver can be demodulated correctly if the signal-to-noise ratio(snr) is above a certain threshold (say T SNR ). Consider the following scenario. A transmitter T and receiver R are separated by distance d, while another sender S (potentially causing interference) is at a distance r from the receiver R. Let P r denote the receiving power of the signal from the transmitter T, and P i the power of the signal from S at R. Thus, the SNR is given as SNR = ambient or thermal noise in the environment around the receiver R. Neglecting N a compared to P i, and assuming homogeneous radios, we have SNR = P r /P i = Pr P i+n a, where N a is the ( ) k r T SNR (2) d r k T SNR.d (3) Thus to successfully receive a signal, the interfering nodes must be at least k T SNR.d distance away from the receiver. This is defined as the interference range R i of the receiver with a transmitter-receiver distance of d, i.e., R i = k T SNR.d. Let us define k T SNR to be the interference factor (α). For the two ray ground model [6] it can be shown that α = Based on this observations we consider the following two ranges for any given transmitter, T : Transmission Range (R tx (T )): This represents the range within which a receiver can successfully receive data transmitted by T if there is no interference from other transmitters. The transmission range is mainly determined by transmission power used by T, and radio propagation properties of the medium. Interference Range(R i (T )): This is the range within which transmission by T will cause interference to receivers that are trying to receive data from other transmitters. The receivers within this range will experience packet losses if T is transmitting simultaneously with their respective sources. In general, R i (T ) is greater than R tx (T ). 1) Interference Range of a BSS (R BSS ): This is the range within which stations will be interfered by either the access point creating the BSS or another station associated to the access point. Scenarios of interference: For simplicity, we assume an open-space environment with circular transmission ranges, etc. In Figure 3, the innermost circle around an AP indicates it transmission radius (i.e. R tx (AP )). We call this radius, R. A wireless user must be located within this transmission radius in order to associate with the AP. The next larger circle indicates the maximum transmission radius among all mobile users associated with this AP (hence the mobile user is located within the innermost circle). The radius of this circle is 2R. Finally the outermost circle indicates the maximum interference radius of the mobile users, which are associated with the AP, and hence is (1 + α)r. For the open-space model, this range is ( )R, i.e. 2.78R. Since each AP and all the mobile users together define a Basic Service Set (BSS) in the terminology, we call this range the interference range of a BSS, denoted by R BSS. We distinguish between the following scenarios as shown in Figure 3, with two APs, (AP 1 and AP 2 ) and their clients (C 1 and C 2 ). Client C 1 is associated with AP 1 and client C 2 is associated with AP 2. Assume that both APs have been assigned to the same channel. No conflict: In this scenario (Figure 3(a)) the two APs are separated by a distance greater than R BSS +R, i.e., 3.78R in the open-space model. In this case the mobile clients of the two APs do not interfere with each other. If the distance between APs is less than or equal to R BSS +R, then transmissions made by a client of one AP will potentially interfere with transmissions of the other AP or its clients. Hence in such cases these two APs should be assigned to different channels for improved user performance. We discuss three such cases next. Type-1 conflict: In this case (Figure 3(b)) the two APs are separated by a distance, D, such that R BSS < D R BSS + R. This implies that transmissions of clients C 1

5 and C 2 interfere with each other. However, because the distance between an AP and clients of the other AP is greater than the interference range of such clients, the AP itself does not observe any such interference. Type-2 conflict: In this case (Figure 3(c)) the distance between the two APs is such that 2R < D R BSS. In this case the client C 1 will interfere at AP 2 and client C 2 will interfere with AP 1. The clients themselves will also interfere with each other (in fact the clients can potentially demodulate the transmissions of each other correctly and hence realize the conflict). However, AP 1 cannot correctly demodulate the transmissions of C 2, since the AP 1 is within the interference range of C 2 but not within its transmission range. Hence neither AP can accurately detect that this interference is due to another another AP located in its vicinity and is using the same channel. Type-3 conflict: In this case (Figure 3(d)) the distance between the two APs is less than 2R. In this scenario, AP 1 will be able to correctly demodulate transmissions by C 2 and will realize that there is another AP in its vicinity using the same channel, i.e. will be able to detect the conflict. Among these four scenarios, the two APs should be allocated the same channel only in the No-conflict case. In the other three cases, the two APs should be assigned to different nonoverlapping channels, if possible. However, out of the three conflict cases shown in the figure, 3, an AP can autonomously detect the presence of Type-3 conflicts only when using the channel allocation scheme. The other two conflict scenarios will go undetected. However, the frequency of Type-1 and Type-2 conflicts in real-life scenarios is expected to be quite high. R BSS C1 2R AP 1 No affected clients in intervening region R AP 2 D [ 2R to R BSS ] Fig. 4. Based on the impact on mobile users, some of the conflict scenarios may not require the APs to use independent channels. Weight of Conflicts: Note that channel re-assignment for Type-1, Type-2, and Type-3 conflicts are necessary if and only if there is at least one client who is affected by the co-channel interference. It is possible that neither of these two APs will have any client in the intervening zone between them. We show such an example in Figure 4 which classifies as a Type-2 conflict. Note that in this scenario, the APs themselves do not interfere with each other (since their distance is > 2R. Their clients are not in the intervening physical space between each other and hence are also not affected if the two APs use the same C2 channel. Clearly in such a scenario, no channel re-assignment is necessary and hence it is important to capture the impact of a conflict by the number of affected users. Hence in Section III we will define our graph theoretic formulation of the problem in which the impact of potential conflicts are captured using appropriate weights. Received Signal Strength Adjacent Channel Interference Receiver channel signal strentgh Fig. 5. The signal strength at the receiver with the transmitter at channel 6. Each point is an average of atleast 1 samples. 2) Exploiting Variations in Adjacent Channel Interference: Adjacent channel interference refers to the interference caused on a particular channel by a station transmitting on a neighboring channel. Two channels are independent or non-overlapping if there is negligible interference between stations operating simultaneously on the two channels. Adjacent channel interference is present in IEEE a/b/g networks. To better understand such adjacent channel interference we performed a simple experiment with IEEE 82.11b. A transmitting station is placed on channel 6, and a receiving station is moved from channel 1 through 11. Figure 5 shows a plot of the signal strength at the receiving stations for the different receiving channel choices. We can see that the interference between channels 1 and 6, and channels 6 and 11 are very low. Hence channels 1, 6, and 11 are considered non-overlapping for all practical purposes. Ideally we will like to use such nonoverlapping channels to avoid any conflict in the WLAN environment. However, there exists other channel pairs where the interference is fairly low. For example, channels 2 and 6, and channels 6 and 1. Although the interference on these channels pairs are greater than that between 1, 6, or 11, such interference can still be considered low for many practical scenarios. We believe that with the steady growth in deployment of APs within the spectrum-limited wireless environment, it will be important to exploit such opportunities of low interference when necessary to achieve greater spatial re-use. We will now define a graph theoretic formulation of our problem where we present a weighted variant of the traditional graph coloring problem, in which the weights will capture the impact of co-channel interference on mobile users, as well as the impact of interference between (potentially) overlapping channels on these users. III. GRAPH THEORETIC FORMULATION In this section, we present a graph theoretic model to capture constraints on channel allocation among interfering access points.

6 A. Overview The channel assignment problem for WLANs can be modeled as a graph coloring problem in which the APs are the vertices of a graph. A conflict between two APs (due to physical proximity and potential interference) is represented by an edge in the graph. The goal of this graph coloring problem is to assign a set of distinct colors (one corresponding to each available channel). To enable the most efficient re-use of these channels the objective of this problem will be to color the graph with the minimum number of colors, a problem which is known to be NP-Hard for general graphs. We call this model, the unweighted graph coloring problem. Note that in the unweighted problem, the colors correspond to non-overlapping channels. As the number of APs within a given physical region grows, the number of colors (nonoverlapping channels) needed to color the graph will exceed the number of available colors (i.e., the maximum number of non-overlapping channels). To enable an efficient channel allocation under such circumstances we extend the above graph theoretic formulation to a weighted graph coloring problem with a certain objective function. In this weighted variant, each vertex corresponds to a distinct AP as before. However, each edge on this graph now has a weight associated with it. The weight of an edge indicates the importance of using different colors (channels) for the corresponding vertices (APs) that are connected by that edge. In the rest of this discussion we will use APs and vertices and similarly colors and channels interchangeably. One informal heuristic for assigning weights to APs can be the total number of active clients, associated to these two APs that will interfere with each other, if both these APs are assigned the same channel. We will discuss the exact computation of these weights in Section IV-C. If two APs are connected by an edge with a high weight, we will want to assign different channels to them. If two such vertices are assigned non-overlapping channels, then the contribution of this edge to the objective function is zero (since the clients of these two APs do not interfere). However, our discussion in Section II indicates that it is possible to assign partially overlapping channels to such neighboring APs. In such a scenario, this edge contributes a positive value to the objective function, e.g., it can be the number of clients affected by interference, scaled by the degree of interference between the chosen channels. Clearly if the two APs are assigned to the same channel, the contribution of this edge to the objective function is maximum, and decreases with decreasing overlap between the assigned channels. Consequently the goal of algorithms solving the weighted graph coloring algorithm is to minimize this objective function, thereby minimizing the impact of interference on clients. We discuss three specific objective functions for this minimum weighted graph coloring problem, and show that the problem is NP-hard for all three. Note that the graph model presented in this section and the techniques presented in section IV can be applied to any set of APs, sharing the same RF spectrum in the region of interest and belonging to potentially different WLANs with arbitrary topologies. B. Formal definition Let k denote the total number of non-overlapping channels available in the underlying wireless PHY layer 3. Given a region of interest covered with a set of access points, define an overlap graph G = (V, E) as follows: V = {ap 1, ap 2,..., ap n } be the set of n APs that form the network. Place an edge between APs ap i and ap j (ap i ap j ) if there is an overlap in the interference region of the BSS created by APs according to the interference model. Let W be the weight function on G; W (ap i, ap j ) denotes the normalized weight on the edge (ap i, ap j ). In this discussion we assume that the weight of an edge indicates the number of clients associated with the two corresponding APs that are affected if these APs are assigned the same channel. The weighted graph coloring problem for channel allocation is stated as follows: A channel assignment C(ap i ), ap i V is a mapping C : V {1... k} from the set of vertices to the set of colors. We say that an edge (ap i, ap j ) is conflict free edge if the interference between these two channels is zero (e.g. channels 1 and 6 in 82.11b). Else if the choice of colors have some positive interference (e.g. say channels 1 and 2 in 82.11b) we call the edge (ap i, ap j ) a conflict edge. We define a term Interference-factor or I-factor, denoted by I ( ap i, ap j ), for each edge, which is the interference between the colors assigned to the two APs. Note that the I-factor of all conflict-free edges is zero. We call the product, W (ap i, ap j ) I(ap i, ap j ) as the I-value. The I-value represents the total effect of interference on all clients that fall in an overlapping region between two APs. We informally call this the conflict edge weight. Thus, given G and W as defined above, the channel allocation problem is to find a mapping C such that an objective function is optimized. Below, we define three different objective functions L max, L sum, L num, which we shall also use as metrics to evaluate the distributed channel allocation algorithms presented in section IV. 1) Minimize the I-value among all interfering APs: Minimize : L max (G, C) = Max I(ap i, ap j )W (ap i, ap j ) e=(ap i,ap j) E We call the problem of minimizing L max, as defined above, a min-max graph coloring problem. This minimizes the maximum I-value of an AP in the graph. Informally this implies that we are minimizing the maximum impact of interference among all overlap regions between APs. 2) Minimize the sum of weights on all conflict edges: Minimize : L sum (G, C) = I(ap i, ap j )W (ap i, ap j ) e=(ap i,ap j) E L sum, as defined above, minimizes the sum of the I- factors. This informally translates to minimizing the total 3 Equal to 3 for IEEE 82.11b/g and 13 for IEEE 82.11a.

7 effect of all interference experienced by clients as a consequence of channel assignments. 3) Minimize the number of conflict edges: L num (G, C) = I(ap i, ap j ) e=(ap i,ap j) E L num, as defined above, minimizes the total impact of all conflict edges, and hence translates to minimizing the total effect of interfering APs. Non-overlapping Channels: Special Case: Note that the non-overlapping channel assignment problem is a special case of our problem formulation above. It can be obtained by defining the I-factor as I(ap i, ap i ) = 1, I(ap i, ap j ) =, j i, and restricting the choice of channels to the corresponding nonoverlapping channels. Although all three objective functions translate to meaningful optimizations, we focus on the L max objective function as it minimizes the maximum I-value among all overlap regions and hence maximizes available bandwidth on a per-channel basis. Below, we show that the above problem is NP-hard for the L max objective function. On similar lines, one can show NPhardness for the other objective functions. Theorem: Given a graph weighted undirected graph G = (V, E), with n vertices, V = {v 1, v 2,..., v n }, and a weight function W : E {1... 1}. Let C : V {1... k} denote a k-coloring of G. The problem of finding P that minimizes L G = Max e=(v V W (v i, v j ) i,v j) E C(v i)=c(v j) is NP-hard. Proof: Consider the general problem of coloring an undirected graph: Given a graph G = (V, E), does there exist a k-coloring C : v {1... k}, 1 k V, such that e = (v i, v j ) E C(v i ) C(v j ). By assigning a weight of 1 to all edges, we formulate this as a min-max coloring problem. Let P be the solution obtained for the min-max version, it can be seen that L G = iff G has a k-coloring. Thus the result follows from the hardness of the general graph coloring problem. C. Centralized and Distributed Solutions Since the unweighted graph coloring problem is known to be NP-Hard, a number of techniques have been proposed in the literature to approximately solve the unweighted graph coloring problem [7]. In the appendix we show how any such approximation algorithm can be adapted to provide a solution to the weighted version of the problem. The focus of this paper is to define distributed techniques to solve our desired problem. We discuss two such distributed solutions next. IV. DISTRIBUTED CHANNEL ALLOCATION ALGORITHMS In this paper we primarily focus on the L max objective function and we present two different heuristics which we call and Hsum to minimize this objective function. Both algorithms are executed at APs in a distributed manner. Both algorithms work on local information (i.e. each AP collects information from its neighbors) and are hence scalable in the size of the wireless network. The algorithms are incremental in nature, hence they can adapt the channel assignment to reflect changes to the graph topology. The first algorithm,, does not require communication between the APs, and applies to multiple co-existing wireless networks sharing a limited RF spectrum. It primarily attempts to reduce the L max objective function. The second algorithm, Hsum, tries to reduce the L max objective function (like the ) algorithm, but additionally is also able to reduce the L sum objective function to a significantly lower value than what is able to achieve, without compromising the L max objective function. To do this, the Hsum algorithm requires some coordination between APs. It uses this coordination function to operate more intelligently than the basic algorithm. Such coordination can be implemented using the Inter-Access Point Protocol [8]. The motivation behind these heuristics is their applicability to non-cooperative APs. We assume that the APs are nonmalicious in nature, but the first technique, does not require collaboration among the APs each AP attempts to maximize its gain by reducing interference locally. The second heuristic improves on the first, however requires some collaboration to achieve better performance for the network as a whole. The algorithms achieve scalability in the size of the network by operating in a local neighborhood. They are also incremental in nature, and keep the channel assignment updated as the network topology changes. The current state of each algorithm is the current color assignment of all APs. Since no color assignment is invalid, the algorithm does not reach any inconsistent state because of packet loss or node failures. The effect of failures would be reflected in a delayed convergence to a better channel assignment. This makes the algorithms fault tolerant in nature. We describe the algorithms using the notation presented in Section III. We define the following additional terms: k is the number of available channels. N(ap i ) denotes the set of neighbors of ap i, i.e. N(ap i ) = {ap j e = (ap i, ap j ) E ap i ap j }. For simplicity of presentation, we define the following objective functions, corresponding to L max and L sum respectively, which are local to an AP ap i : L max (G, C, ap i ) = Max I(ap i, ap j )W (ap i, ap j ) e=(ap i,ap j) E (4) Informally, L max (G, C, ap i ) is the maximum conflict weight of an edge incident on ap i. L sum (G, C, ap i ) = I(ap i, ap j )W (ap i, ap j ) e=(ap i,ap j) E (5) Informally, L sum (G, C, ap i ) is the aggregate of all weights of conflict edges incident on ap i. A. Algorithm : The algorithm, outlined in Figure 6, has two steps discussed in detail below:

8 Procedure : (ap i ) 1. Initialize: C(ap i ) 1 { initial coloring for all nodes } 2. Optimize: (a) H(c) Max I(ap i, ap j )W (ap i, ap j ) e=(ap i,ap j) E C(ap j)=c { H(c) is max. weight of any edge to ap i whose neighbor has a color c } (b) Choose c such that H(c ) = Min c=1...k H(c) { Choose color with min. conflict weight } (c) C(ap i ) c Fig. 6. Algorithm. Initialization step: This step selects an initial assignment for C. This can be as simple as assigning the same color to all APs, or the APs can perform a Least Congested Channel Search () (discussed in section II) and leverage that as an initial coloring. We study the effect of different initial colorings on L max in section V. Optimization step: This step improves the coloring in an incremental manner. Each AP ap i attempts to minimize interference on its maximum conflict edge by selecting the color which minimizes L max (G, C, ap i ). This is done in steps 2(a) and 2(b) of the algorithm. The optimization step is executed by each AP periodically. This moves the network to a better coloring and keeps the channel assignment in synchrony with changes to the graph topology. Changing the channel of operation: The above algorithm executes on an AP after the wireless network has started providing link layer connectivity. Hence changing an APs channel of operation is costly as it requires all clients to dis-associate and re-associate again, suffering a temporary loss of service. This can be taken into consideration in the following manner: 1) In presence of Coordination between APs: If the APs can communicate with each other, for example using the Inter-Access Point Protocol (IAPP) [8], the optimization phase can be executed on a virtual-color variable maintained at each AP. Once a sufficiently 4 better color assignment has been obtained, the operation of changing channels can be performed during periods of low activity. 2) In absence of Coordination between APs: If the APs cannot communicate with each other, they have to change their channel of operation during each execution of the optimization step. Being greedy in nature, an execution of the optimization step will only reduce the interference. Since each execution step has to be performed during periods of low network activity, the algorithm may take a longer time to converge on the overall. Thus the tradeoff for lack of coordination is an increased convergence time to a better channel assignment. However, when compared to the duration of service offered by an AP 5, this time 4 Any of the metrics L max, L sum, L num can be used to compare two color assignments. 5 Which can be assumed to be of the order of days or months on average. would still be negligible. B. Algorithm Hsum: We now present the Hsum algorithm which performs equally well (as ) in reducing the Lmax objective function, but additionally is also able to reduce the L sum objective function without compromising the Lmax objective. Hsum is able to perform such an improved channel assignment by requiring some coordination between participating APs to communicate the L max value during the execution of the algorithm. Hence, this algorithm particularly applies to wireless networks under the same administrative domain. The algorithm can also be applied to situations where APs from different WLANs are willing to coordinate and communicate with each. Such a communication can be achieved using the IAPP [8] protocol. Due to space constraints, in this paper we just provide a high level description of this algorithm. Pseudo-code is outline in Figure 7. The previous algorithm, each AP performs a a greedy optimization trying to minimize its local maximum interference, i.e. minimize L max (G, C, ap i ). Although this reduces the maximum interference for each AP, it can create many conflict edges with low weights, which can lead to potentially large aggregate interference in the entire network, i.e. L sum. Hence Hsum tries to additionally reduce the L sum objective function as well as follows: It minimizes L max (G, C, ap i ) if L max (G, C) = L max (G, C, ap i ). In other words, if the AP ap i has a conflict edge which has the maximum weight over the entire graph, it selects a coloring which minimizes L max, otherwise it minimizes the the aggregate interference within its area of coverage, i.e., L sum (G, C, ap i ). IAPP [8] based coordination is necessary for this algorithm because each AP needs to periodically evaluate the global value of L max. Note that since IAPP operates at the network layer, its enables APs from different administrative domains/wlans to communicate with each other. The IAPP messages would typically be exchanged using the wired Distribution System connecting the APs, and thus the overheads due to this communication are not relevant. For restricting the assignment to only non-overlapping channels, as a special case, the I-factor can be set to I(ap i, ap i ) = 1, and I(ap i, ap j ) = in the above algorithm. This has been discussed in section III. C. Distributed Construction of Overlap Graph An important input to the channel assignment algorithms is the overlap graph construction. A naive method would miss the Type-1 and Type-2 conflicts and hence we need to handle theses cases in the construction methods. We present two scalable and autonomous methods of constructing the overlap graph in a distributed manner. The first method is based on a new technique included in the IEEE 82.11K Radio Resource Management draft [9] called the Site-Report and will be supported by wireless cards in the future. Current wireless cards can also easily support the site-report feature through firmware updates. Although the site report method is preferred for generating the overlap graph, we discuss another method based on

9 Procedure : Hsum(ap i ) 1. Initialize: C(ap i ) 1 { initial coloring for all nodes } 2. Optimize: { Define w max as current value of L max (C). } (a) H(c) Max I(ap i, ap j )W (ap i, ap j ) e=(ap i,ap j) E C(ap j)=c (b) if H(c) w max M(c) 1 else M(c) { Mark colors with the max. conflict weight > w max We do not want to choose such colors. } (c) S(c) e=(ap i,ap j) E C(ap j)=c I(ap i, ap j )W (ap i, ap j ) { S(c) denotes sum of weights of all edges to ap i whose neighbor has a color c } (d) Choose c such that S(c ) = Min S(c) 1 c k M(c)= { Choose color with min. sum of conflict weights among all unmarked colors. } (e) C(ap i ) c Fig. 7. Algorithm Hsum. AP to which the client was associated prior to the reassociation. An edge is added between two APs upon the receipt of this message. Since most of the reassociations happen due to degrading signal strength [11], it can be assumed that the client was on the boundary of the coverage area of the prior AP. Hence the signal strength of the reassociation message received by an AP would give a direct indication of the weight on the edge between this AP and the prior AP. In cases where there is a significant overlap between two APs, it might not be possible for a client to perform a reassociation between them. In such cases, the edge would go undetected by the neighbor graph technique. Here we enhance this technique to include the detection of such edges. Called the channel-scan enhancement, each AP performs a periodic scan of all channels to capture the significant overlap edges. The scan can be performed less frequently and during periods of low network activity. The overlap graph can hence be constructed using the channel-scan enhancement along with the neighbor graph technique. In section VI, we demonstrate the construction of an overlap graph using both methods for an in-building wireless network testbed. capturing IEEE reassociations and channel scans. This method works with the current hardware. Both methods update the graph incrementally and on a periodic basis, hence incorrectly added edges can be removed over time and changes to the network topology will get reflected into the graph over time. 1) Using Site-Reports: An AP can randomly and with very low probability, request a client to perform a site-report during intermittent periods of low activity with respect to the client. This requires the client to perform a passive scan of each channel and reports a list of APs along with the received signal strength. An edge is placed between the current AP and any APs reported by the client. The weight of an edge created using the above method can be calculated as follows. Let Num api denote the number of site reports conducted by AP ap i. Let Numap i (ap j ) denote the number of site reports conducted by AP ap i which reported ap j. Then the weight W (ap i, ap j ) on the edge (ap i, ap j ) is obtained by W (ap i, ap j ) = Numap i (apj) Num api. Since the site-report is performed by clients associated to and within the range of an AP, this weight indicates the fraction of clients associated to ap i that are affected by interference from ap j. Thus higher the weight, greater would be number of clients affected and hence the channel assignment algorithms would assign nonoverlapping or partially overlapping channels to these APs. The weight can be calculated by an AP without any communication with other APs. In the case that the APs coordinate with each other, a better statistical estimate for W (ap i, ap j ) would be Numap i (apj)+numap j (api) Num api +Num apj. 2) Using Reassociations and Channel Scans: In [1], Mishra et. al. present a technique called Neighbor Graphs which captures the reassociaton relationship between APs. During a reassociation, the client sends an IEEE reassociation request message which contains the MAC address of the V. SIMULATIONS We have evaluated the performance of our proposed algorithms through extensive simulations as well as through measurements on an operation wireless testbed, consisting of 2 APs distributed on two floors. We present the results from our simulation study in this section, and the results of our measurement study on the wireless testbed in the next section. A. Methodology The goal of the simulation study was (1) to understand the performance of our proposed techniques in a controlled manner, and (2) to study its scalability properties with increasing number and density of APs in the environment. In this study we assumed that the APs were distributed over a three dimensional physical region. The locations of the APs were chosen uniformly at random for the different experiments. Each AP had a transmission radii and interference radii which was Additionally we assumed that the clients were distributed uniformly at random in the same physical space. Each client associated with a nearby AP with the strongest signal strength. Each client in this topology is an active client, i.e., it will cause and experience interference with other clients as they communicate with their respective APs. Each client also has its independent transmission and interference radii. Clearly the choice of circular radii for transmission and interference ranges is a simplifying assumption and we study the impact of more realistic physical wireless environments in the next section. In all these experiments we compared the performance of three channel allocation algorithms, namely the current best known technique as defined in Section II, and our proposed distributed algorithms and Hsum. The actual channel selected by the algorithms depends on the channel conditions, the number of clients associated to neighboring APs,

10 Maximum Conflict Weight(L-Max) Maximum Conflict Weight (L-Max) with respect to 1 Hsum Sum of Weights(L-Sum) on conflict edges Sum of Weights(L-Sum) with respect to Hsum Number of Conflict Edges(L-num) Number of Conflict Edges(L-num) with respect to 2 Hsum (a), Hsum, with L max (b), Hsum, with L sum (c), Hsum, with L num Fig. 8. Figure compares the m, Hsum and with the number of colors. Each point is an average of simulations over 1 different graphs each with 25 vertices, a maximum degree of 8. Maximum Conflict Weight(L-Max) Maximum Conflict Weight(L-Max) with respect to 1 Hsum Sum of weights on Conflict Edges(L-Sum) Sum of Weights(L-Sum) with respect to Hsum Number of Conflict Edges(L-Num) Number of Conflict Edges(L-Num) with respect to 4 Hsum (a), Hsum, with L max (b), Hsum, with L sum (c), Hsum, with L num Fig. 9. Figure compares the m, Hsum and with the number of colors. Each point is an average of simulations over 1 different graphs each with 1 vertices, and a maximum degree of 2. and the network traffic going through the interfering APs. We capture this dynamic nature of the algorithm in the simulations. Performance Metrics: We primarily examined three performance metrics for this study, as given by the three objective functions, L max, L sum, and L num, that we defined in Section III. Note that in any wireless environment it is always desirable to minimize each of these metrics. Simulation Scenarios: We present different simulation scenarios. Our first results in this section will examine the channel assignment properties of the algorithms if the objective is to only assign non-overlapping channels to APs. This is the current practice used for channel assignment today. Such a nonoverlapping channel assignment problem is a special case of our problem formulation, and can be handled by both and Hsum algorithms by appropriately choosing the I-factor, i.e., by setting I(ap i, ap i ) = 1 and I(ap i, ap j ) =, j i, and restricting the choice of channels to precisely the set of non-overlapping channels. We simulate two different wireless environments one with small(er) number of APs and lower consequently lower interference, and the other with a high(er) number of APs and consequently greater interference. Subsequently we will discuss the impact of increasing interference of the quality of solutions achieved by the different algorithms. Finally we describe the performance of the algorithms when it is permissible to use overlapping channels. Clearly such an approach will lead to better spatial re-use. B. Channel Allocation Algorithms 1) Non-overlapping channels and Low Interference: We consider a wireless environment in which there are 25 APs distributed uniformly at random. The low interference was simulated by bounding the maximum transmission and interference radii of APs and clients. In this example, this typically bounded the maximum number of conflict edges for any AP to 8, i.e., the maximum degree on the overlap graph was 8. Clearly such a topology can always be colored with 9 colors. Hence we varied the number of non-overlapping channels between 3 and 9. Figure 8 compares the three algorithms. 1) Both and Hsum outperform the algorithm. As the number of colors is increased from 3 to 7, the performance difference between the, Hsum and the algorithm increases. In fact, even as the number of colors increase, does not improve as much, because it cannot detect many conflicts that exist, e.g., the Type-1 and Type-2 conflicts. The plot is restricted to a maximum X-value of 7 channels because both and Hsum was able to find perfectly conflict-free channel allocation when using 7 nonoverlapping channels. Such a channel assignment cannot be obtained by performing an. 2) Hsum does better than. This is because Hsum tries to minimize the L sum metric in additional to

11 L max using additional coordination between APs using IAPP [8]. 2) Non-overlapping channels and High Interference: To simulate higher interference in the wireless environment we use a larger number of APs to be located in the same physical space as before with same transmission and interference ranges. The number of randomly distributed APs in this case was 1, which led to a maximum degree of 2 in consequent the overlap graph. This implied that the graph is definitely colorable without conflicts using 21 colors. Our results will show that and Hsum could achieve conflict-free channel assignments to APs by using only 16 non-overlapping channels. In Figure 9 we present results comparing all the three algorithms in this environment. We can make the following observations: 1) and Hsum exhibit similar improvements as with the smaller graph of figure 8, irrespective of the dense connectivity in the graph. However, the algorithm does much worse than in the previous simulations. As interference in the environment increases, is unable to detect an even larger number of Type- 2 and Type-3 conflicts. With the proliferation of WLANs today, such increased interference in the wireless environment will be very commonplace today and in the future, underlying the need for efficient channel assignment techniques like and Hsum. 2) Like before Hsum does better than with respect to L sum and L num and the two algorithms performs almost equally well with respect to L max. Non-overlapped channels and Effect of Increased Interference: Comparing the performance of, Hsum, and above, we observe that the performance gains of our distributed algorithms increase with increasing interference. We now explore the impact of such increasing interference by explicitly increasing the degree of APs in the overlap graph. (This is equivalent to increasing interference.) Figure 1(a) and (b) presents the performance of the and Hsum algorithms in identical environments where we varied the maximum degree of an AP on the overlap graph. In all such overlap graphs, the average degree was within 15% of the maximum degree in these experiments. The performance of all algorithms degrades with increasing degree, but as the number of colors increases Hsum does significantly better than. C. Non-overlapping Channels with Multiple Co-existing Wireless Networks We now study the impact of the proposed algorithms on multiple co-existing wireless networks within the same physical environment, and hence need to share a specific set of available channels. We assume in these experiments that these networks are greedy but not malicious, i.e., if a good channel is available an AP will try to move into that channel. However, as a channel gets congested with multiple APs, they will try to migrate to other less congested channel, either by using our proposed techniques or the method 6. If all the APs implement either 6 A malicious AP can send jamming signals on a specific channel to drive away other APs from prior to acquiring the channel. We do not model such behavior in this paper. Handling such malicious behavior (like physical jamming) is usually difficult in the unlicensed spectrum. the and Hsum algorithms, it does not matter whether they belong to the same network or different networks. In either case there will be significant performance improvements as described. In this section, we examine the interesting scenario where the different networks employ different channel allocation algorithms specifically one network employs the algorithm for channel assignment while the other uses. We define a parameter λ as the percentage of the APs executing to the total number of nodes. The remaining nodes implement the algorithm. The simulation results presented earlier have λ = 1%. Figure 11 (a)-(c) shows the simulation results on topologies with 1 APs, and an interference level such that the maximum degree on the overlap graph is 15. We demonstrate results for five different λ parameters, λ =, 1, 25, 5, 1. The curve indicates λ =, while the curve indicates λ = 1. Figure 11 (a) plots the L max objective function for the different scenarios, Figure 11 (b) plots L sum, and Figure 11 (c) plots L num. In the L max plot (Figure 11 (a)), the lines for λ = 1, 25, 5 practically overlap each other and hence we plot just one of them. We can observe the following: 1) As the value of λ increases from to 1, the curves move closer to. Thus more the ratio of nodes executing, better would be the performance of the wireless network as a whole. 2) Interestingly, even if only a small percentage of the nodes execute, it still improves the metric drastically for the network on the whole. This happens particularly if the number of colors available is small. For example, in Figure 11(b), for λ = 1%, i.e., 1% of the nodes execute, there is a 25% reduction in the value of L sum (over just ) for 3 colors and a 4% reduction in its value for 4 colors (over just ). D. Assigning Overlapping Channels We now present results for the channel assignment problem when we are allowed to assign partially overlapped channel. As we discussed, this flexibility enables greater spatial reuse in the wireless environment. In these simulations we assume that the number of non-overlapping channels is three, and the total number of channels is 11 (corresponding to 82.11b). For the simulations we empirically measured the I-factor, which defines the impact of assigning partially overlapping channels to neighboring APs, for all possible channel assignment pairs. We had presented the relative I-factor between channel 6 and all other channels in Figure 5. We present the results for these experiments in Figures12. Note that when using overlapping channel assignments, we refer to our algorithms as ADJ-minmax and ADJ-sum instead of and Hsum respectively. In the plots we compare the performance achieved by assigning overlapping channels to that achieved by assigning only non-overlapping channels. Clearly the performance achieved by assigning overlapping channels is expected to be better (i.e. ADJ-minmax is expected to achieve better performance than. Figure 12(a) compares the performance of ADJminmax, and. On the whole, ADJ-minmax

12 1 Maximum Conflict Weight(L-Max) with respect to Max-degree 9 Maximum Conflict Weight (L-Max) with respect to Max-Degree Maximum Conflict Weight (L-Max) colors 4 colors 5 colors 6 colors Maximum Degree of a vertex Maximum Conflict Weight (L-Max) colors 4 colors 1 5 colors 6 colors Maximum degree of a vertex (a) with respect to L max (b) Hsum with respect to L max Fig. 1. Figure (a) compares and Hsum with respect to the maximum degree of a vertex. Figure (b) compares and Hsum with respect to the number of vertices keeping the maximum degree = 6. Each point is an average of 5 different graphs with the same parameters. Maximum Conflict Weight (L-Max) Maximum Conflict Weight (L-Max) Vs - complete graph - Lambda = 5 Sum of Conflict Weights (L-Sum) Sum of Conflict Weights (L-Sum) Vs - complete graph - Lambda = 5 - Lambda = 25 - Lambda = 1 Number of Conflicts Number of Conflicts (L-Num) Vs - complete graph - Lambda = 5 - Lambda = 25 - Lambda = Number of colors Number of colors (a), (λ = 5), with L max (b), (λ = 1, 25, 5), with L sum (c), (λ = 1, 25, 5), with L num Fig. 11. Figure compares with various Lambda values and. Each point is an average over 1 graphs, each with 1 vertices and a maximum degree of Maximum Conflict Weight (L-Max) Vs Number of Vertices 1 Maximum Conflict Weight (L-Max) Vs Number of Vertices Maximum Conflict Weight (L-Max) ADJ-minmax Number of vertices Maximum Conflict Weight (L-Max) ADJ-sum Hsum Number of vertices (a), ADJ-minmax and with L max (b) Hsum, ADJ-sum and with L max Fig. 12. Figures shown above compare, Hsum, ADJ-minmax, ADJ-sum and with the number of vertices. Each point is an average of simulations over 1 different graphs with the same parameters. improves the L max objective function. Specifically, as the number of vertices increase, ADJ-minmax does increasingly better than. The lack of non-overlapping channels causes to suffer, while ADJ-minmax utilizes partially overlapping channels to improve the objective function. Figure 12(b) shows the performance of ADJ-sum Hsum and. Like ADJ-minmax ADJ-sum outperforms Hsum, and the performance difference increases with the number of APs. Among themselves, ADJ-sum does better than ADJ-minmax with respect to L sum and L num and the two algorithms performs almost equally well with respect to L max the simulations are shown in a technical report. Convergence Results: Both algorithms and Hsum converge rapidly. In particular, converges in 2 rounds of execution of the optimization step where one round is a single execution on each AP in the topology. Hsum converges in

13 4 rounds on average. A detailed simulation based analysis is presented in a technical report. VI. EXPERIMENTS BASED ON TESTBED MEASUREMENTS 3rd FL 2nd FL Fig. 13. Figure shows the overlap graph for the testbed network. The solid lines show the edges in the neighbor graph, and the dashed lines show the remaining edges. The number in the circle for each vertex indicates the current channel assignment in use which was performed using the algorithm. Fig. 14. Figure shows performance improvements achieved by and ADJ-minmax over the current channel assignments used by the network with respect to the L max metric. Maximum Conflict Weight(L-Max) Maximum Conflict Weight (L-Max) with respect to Fig. 15. Figure shows performance improvements achieved by over with increasing number of colors using uniformly distributed weights over the overlap graph created from the testbed with respect to the L max metric. Based on the measurements performed on an in-building testbed wireless network, we evaluate the channel assignments achieved by the and the algorithms with overlapping channel re-use. Note that all physical effects of an inbuilding environment including path loss and multi-path effects have been captured in our measurements. 1) The Wireless Testbed: The wireless testbed network consists of 2 IEEE 82.11b APs distributed over two floors of an office building. The APs are Soekris boards [12] with a IEEE b Prism II wireless card configured as a host-based AP. All APs in the testbed operate at 1mW of transmit power. The initial channel assignment on the APs was done by searching for the least congested channel. 2) Computation of the Overlap Graph: Figure 13 shows the overlap graph computed for the testbed network. The overlap graph was generated using the two methods outlined earlier in section IV-C, namely the Site-Report method and the Neighbor Graphs method with the channel-scan enhancement. Using Site-Report: Since the site-report method outlined in the IEEE 82.11K draft [9], is currently not implemented in the firmware, we performed the site report by using a dedicated laptop as a sniffer. The sniffer machine is an IBM ThinkPad T23 running Linux, with a Prism II wireless card in the monitor mode to capture raw IEEE frames with signal strength information. Using Neighbor Graphs: Here an edge is placed between two APs if a client performs a re-association between them. The following experiment was performed: A client and a sniffer were made to move randomly within the coverage area of the wireless testbed, and the re-association edges were created. This captures all edges present in the neighbor graph. As discussed in section IV-C, this will not capture edges where two APs have significant overlap. These additional edges were created by sniffing all the channels at each APs location, essentially emulating the channel-scan enhancement discussed in section IV-C. The solid lines in figure 13 show the edges captured by the basic neighbor graph technique. The dashed lines indicate the edges captured by our enhancements to the neighbor graph technique discussed in section IV-C. 3) Experiment Results: Based on the overlap graph inferred as above, we evaluate our channel allocation methods. This is compared with the current channel allocation on the testbed which based on the algorithm. 1) We first present results from assigning non-overlapping channels using. In Figure 15, we compare with the algorithm with an increasing number of non-overlapping channels. The figure shows that dramatically reduces the number of users suffering interference when compared to as the number of non-overlapping channels increase (for example, with IEEE 82.11a). In particular, with 5 nonoverlapping channels, eliminates interference among access points while the users under the algorithm still suffer considerable interference. 2) Second we compare the performance gain using the 11 overlapping channels, when compared to using 3 nonoverlapping channels. Figure 14 shows the performance improvement over the overlap graph inferred for the testbed wireless network. In the figure, ADJ-minmax refers to the execution of with 11 overlapping channels. It can be seen that, which uses 3 nonoverlapping channels, reduces the number of users under interference by 11%(= 1 (8/9)). ADJ-minmax, which uses all the 11 channels reduces the number of users under interference by 4% over the channel assignment, and by 32.5% over the channel assignment obtained from. The drastic improvements stem