{'fraction_non_alphanumeric': 0.04001866439766146, 'fraction_numerical': 0.009496884692449152, 'mean_word_length': 4.542135686035899, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Distributed Resource Allocation for Proportional Fairness in Multi-Band Wireless Systems

A challenging problem in multi-band multi-cell self-organized wireless systems, such as multi-channel Wi-Fi networks, femto/pico cells in 3G/4G cellular networks, and more recent wireless networks over TV white spaces, is of distributed resource allocation. This involves four components: channel selection, client association, channel access, and client scheduling. In this paper, we present a unified framework for jointly addressing the four components with the global system objective of maximizing the clients throughput in a proportionally fair manner. Our formulation allows a natural dissociation of the problem into two sub-parts. We show that the first part, involving channel access and client scheduling, is convex and derive a distributed adaptation procedure for achieving Pareto-optimal solution. For the second part, involving channel selection and client association, we develop a Gibbs-sampler based approach for local adaptation to achieve the global objective, as well as derive fast greedy algorithms from it that achieve good solutions.


I. INTRODUCTION
Many of the existing and evolving wireless systems operate over spectrum that spans multiple bands. These bands may be contiguous, as in OFDM-based systems, such as current IEEE 802.11-based WLANs (a.k.a. Wi-Fi networks) and evolving fourth-generation LTE cellular wireless systems; or they may be spread far apart, as in multi-channel 802.11 systems and in recently proposed wireless broadband networks over TV white spaces (discussed in the sequel). A common issue in these multi-band systems is how to perform resource allocation among different clients, possibly being served by different access points (APs). This needs to be done so as to efficiently utilize wireless resources-spectrum, transmission opportunities and power-while being fair to different clients. Furthermore, unlike traditional enterprise Wi-Fi networks and cellular wireless networks, where the placement of APs and their operating bands are arrived at after careful capacity/coverage planning, more and more of the evolving wireless systems are going to be self-organized networks. There is extensive literature on completely self-organized wireless networks, also referred to as ad hoc networks [27], which are often based on 802.11. Even the emerging 4G cellular wireless networks, such as those based on LTE, are going to have a significant deployment of self-organized subsystems: namely, pico cells and femto cells [10]. These self-organized (sub)systems will require that resource allocation is performed dynamically in a distributed manner and with minimal coordination between different APs and/or clients.
Another emerging scenario is of broadband wireless networks operating over TV white spaces [7], [12]. Recent conversion to all-digital TV broadcast has made available valuable lower-frequency spectrum. A part of this spectrum, referred to as TV white spaces, has been mandated by FCC for unlicensed broadband access. However, designing a networking stack over the large and fragmented TV white spaces, which may have widely different propagation characteristics, pose new technical challenges unseen in traditional wireless networks and call for entirely new wireless design principles.
In this paper, we consider the problem of joint resource allocation across different APs and their clients so as to achieve a global objective of maximizing the system throughput while being fair to users. Unlike the maxmin fairness used in traditional Wi-Fi networks, which has been shown in several extensive studies to be inefficient [18], [26], we will focus on proportional fairness. The latter has become essentially standard across current 3G cellular systems, as well as in emerging 4G systems based on LTE and WiMAX. Thus, the system objective will be to allocate wireless resources, spectrum and transmission opportunities, so as to maximize the (weighted) sum of the log of throughputs of different clients, which is known to achieve (weighted) proportional fairness.
The combined resource allocation in a multi-band multi-cell wireless system involves four components: Channel selection, client association, channel access, and client scheduling. The first component, Channel Selection, decides on how different APs share different bands of the spectrum available to the wireless system. As mentioned earlier, the emphasis will be on self-organized networks, which will necessitate a completely distributed approach that dynamically adapts to varying traffic requirements in different cells. To achieve these objectives, we will consider a differentiated random-access based solution. The second component, Client Association, allows a client to decide on an AP to associate within its neighborhood that is likely to provide the "best" performance. Unlike the traditional approach of associating with the AP having the highest signal strength, we will discuss an approach that allows a client to associate with an AP that maximizes its proportionally-fair throughput while minimizing the impact on other clients. Once an AP has chosen a channel/band to operate in and a bunch of clients have associated with it, the third component, Channel Access, decides when it should access the channel so as to serve its clients while being fair to other access points in its neighborhood operating in the same channel. The final component, Client Scheduling, decides which of its clients an AP should serve whenever it successfully accesses the channel.
Our approach addresses the four components in a unified framework, where the solutions to different components are arrived at through separation of time scales of adaptation. More specifically, our formulation allows for the optimization problem of maximizing the clients throughput with weighted proportional fairness to naturally dissociate into two sub-parts, which are adapted at different time scales. Assuming that the channel selection and client association have been performed, we show that the sub-problem of channel access and client scheduling becomes convex, which is also amenable to a distributed adaptation for achieving Pareto-optimal weighed proportional fairness. At a slower time scale, we adapt the channel selection and client association to varying demands and interference. This part is a non-convex problem in general, and thus, difficult to solve for a globally optimal solution. We develop a Gibbs-sampler based approach to perform local adaptation while improving the global system objective. The adaptation is randomized, and if done slowly enough, can achieve a globally optimal solution. In practice, however, that may not be always feasible; hence, we derive greedy heuristics from it for channel selection and client association, which though not globally optimal, provide fast and good distributed solutions with limited exchange of information, as the simulation results indicate. The paper is organized as follows. Section II provides an overview of some related work. Section III describes the multi-cell multi-band wireless system model and the joint resource allocation problem that we study. Section IV gives an overview of our approach and discusses the separation of problems. The details of our approach and its desirable properties, including convergence to Paretooptimal proportionally-fair allocation, are given in Section V and Section VI. The results of simulations are provided in Section VII. Concluding remarks are discussed in Section VIII.

II. RELATED WORK
There is extensive literature addressing one or a subset of the aforementioned four components of wireless resource allocation in the context of different wireless systems-Wi-Fi networks, 3G/4G cellular networks, and more recent, wireless broadband systems over TV white spaces. Due to space limitation, we only discuss a small sample of the results in each of these areas.
Distributed resource allocation has been widely studied for IEEE 802.11-based systems. A number of CSMAbased random-access approaches have been developed to provide differentiated services to clients [5], [13], [16], [22], [28]. Proportional fairness in multi-contention neighborhood has been studied in [17], [19]. Specifically, [17] has established that Pareto-optimal weighted proportional fairness can be achieved in a distributed manner with minimal exchange of information among contending clients. Our approach for channel access is motivated by [17]; we, however, consider a more general framework that jointly addresses all the four components. Multi-channel MACs for Wi-Fi networks have been proposed in [6], [23]. Approximation algorithms for client association control to achieve proportional fairness have been developed in [8], [21], but they assume that APs do not interfere with each other through a pre-assignment of orthogonal channels. The work closest to this paper is [20] where the authors have developed Gibbs-sampler based distributed algorithms for channel selection and client association. Their approach, however, considers different objectives for the two components, neither of which ensures proportional fairness.
For cellular wireless data systems, a number of centralized approaches for single-cell scheduling to achieve proportional fairness have been developed [3], [4], [9]. More recently, several inter-cell interference coordination (ICIC) techniques have been proposed for interference mitigation in LTE-based 4G cellular networks. Specifically, [24], [25] have developed distributed algorithms for dynamic fractional frequency reuse among interfering macro cells based on limited exchange of interference information over dedicated control links. Given the nonconvexity of the problem, these algorithms aim to achieve a local optimum of the weighted sum of the log of user throughputs, which as mentioned earlier provides proportional fairness, through local estimates of the appropriate gradients. However, in self-organized subsytems of such networks (such as those made of pico and femto cells), explicit exchange of information may not be feasible [2], and thus, these approaches may not be directly applicable.
Research on resource allocation in wireless networks operating over fragmented TV white spaces is in a nascent stage. [7] considers a single AP serving multiple clients at the same rate. For this scenario, it addresses three issues: how the AP chooses a suitable band, how a new client detects the AP's operating band, and how disruptions due to temporal variations, such as caused by wireless microphones, are handled. Some of the limitations of this approach are overcome in [11] by considering a multi-rate multi-radio architecture. It develops a joint strategy for white space selection and client assignment to one of the radios, as well as designs an extension to CSMA to achieve proportional fairness-all for a single-AP scenario. This paper generalizes the setup by considering a system of multiple APs and jointly addressing the four components of resource allocation to achieve the global system objective of maximizing clients throughput in a proportionally fair manner.

III. SYSTEM MODEL
We consider a system with several access points (APs) and clients that can operate in a number of channels. We denote the set of APs by N, the set of clients by I, and the set of channels by C. Each client i is associated with an AP, which we denote by n(i), and is served by that AP. Each AP n is equipped with u n radios that can operate in different channels simultaneously and each client i is equipped with only one radio. When an AP n has more than one radios, i.e, u n > 1, we can simplify the model by assuming that there are u n APs, each with one radio, that are placed at the same place as AP n. Each of these u n APs corresponds to one radio of AP n. This procedure allows us to only consider the model in which each AP has one radio and simplifies notations. Throughout the rest of the paper, we assume that each AP only has one radio and operates in one channel unless otherwise specified. The channel that an AP n is operating in is denoted by c(n). APs can switch the channels that they operate in, although such switches can only be done infrequently due to the large overheads. When an AP switches channels, all its clients also switch channels accordingly.
We focus on a server-centric scheme where each AP schedules all transmissions between itself and all clients that are associated with it. This scheme is applicable to a wide varieties of wireless systems that include LTE, WiMax, and IEEE 802.11 PCF. Later, we will also discuss a distributed scheme where clients contend for service from APs, such as in 802.11 DCF. We assume that time is slotted, with the duration of a time slot equals the time needed for a transmission. If an AP n makes a successful transmission toward client i in channel c, client i receives data at rate of B i,n,c in this slot. Since characteristics of different channels may be different, B i,n,c depends on c.
On the other hand, we assume that the APs are not synchronized and may interfere with each other. We consider the interference relations using the protocol model [15]. When an AP n operates in channel c, it may be interfered by a subset M n,c of APs, where n ∈ M n,c for notational simplicity. When the AP n schedules a transmission between itself and one of its clients, the transmission is successful if n is the only AP among M n,c that transmits in channel c during the time of transmission; otherwise, the transmission suffers from a collision and fails. We assume that the interference relations are symmetric, i.e., m ∈ M n,c if and only if n ∈ M m,c . Note that, since the propagation characteristics of different channels may be different, especially in the case of TV white space access, the subset M n,c depends on the channel c. This dependency further distinguishes our work from most existing works on multi-channel access where interference relations are assumed to be identical for all channels. We also define Since APs are not coordinated, we assume that they access the channel by random access. Each AP n chooses a random access probability, p n . In each slot, AP n accesses the channel c(n) with probability p n . The transmission is successful if n is the only AP in M n,c(n) that transmits over channel c(n). Thus, the probability that AP n successfully accesses the channel can be expressed as (1) The AP is in charge of scheduling transmissions for its clients. When AP n accesses the channel, it schedules the transmission for client i, where n(i) = n, with probability φ i,n , φ i,n ≥ 0 and i:n(i)=n φ i,n = 1. Since the data rate of client i when it is served is B i,n(i),c(n(i)) and the probability that the AP n(i) makes a successful transmission is as in Eq. (1), its throughput per time slot is, assuming n(i) = n and c(n) = c, We now discuss an analog model for a distributed scheme where clients contend for the service from APs, which can be applied to completely distributed scenarios, such as those based on 802.11 DCF. In this scheme, each client i contend for the channel by accessing it with probability p i in each time slot. Two clients, i and j, interfere with each other if their associated APs interfere with each other, that is, c(n(i)) = c(n(j)) and n(i) ∈ M n(j),c(n(j)) . Client i successfully accesses the channel if none of the other clients that can interfere with it access the channel simultaneously. Thus, the long-term throughput per time slot for client i is , assuming n(i) = n and c(n) = c, r i := B i,n,c pi 1−pi j:c(n(j))=c,n(j)∈M n,c (1 − p j ). Finally, we assume that each client is associated with a positive weight w i > 0. We also denote the total weights of clients associated with AP n by w n , i.e., w n := i:n(i)=n w i . The goal is to achieve weighted proportional fairness among clients, that is, to maximize i∈I w i log r i . The solution to the distributed scheme is very similar to that to the server-centric scheme. Thus, we will focus on the server-centric scheme and only report key results of the distributed scheme.
IV. SOLUTION OVERVIEW AND TIME-SCALE SEPARATION We now give an overview of our approach to achieve weighted proportional fairness, which consists of separating the problem into four components and solving them. By Eq. (2), we can formulate the problem of achieving weighted proportional fairness as the following optimization problem: i:n(i)=n φ i,n = 1, for all n. Based on this formulation, the problem of achieving weighted proportional fairness consists of four important components, in increasing order of time scales: First, whenever the AP accesses the channel, it needs to schedule one client for service. That is, the AP has to decide the values of φ i,n . Second, in each time slot, the AP has to decide whether it should access the channel, which consists of determining the values of p n . Third, each client needs to decide which AP it should be associated with, i.e., deciding n(i). Finally, each AP n needs to choose a channel, c(n), to operate in. We denote the four problems as Scheduling Problem, Channel Access Problem, Client Association Problem, and Channel Selection Problem, respectively. Weighted proportional fairness is achieved by jointly solving the four problems. The problem of achieving weighted proportional fairness for the distributed scheme can be formulated similarly and involves three components: the Channel Access Problem, which chooses p i for clients, the Client Association Problem, and the Channel Selection Problem.
Since the overhead for a client to change the AP it is associated with and for an AP to change the channel it operates in are high, solutions to the Client Association Problem and the Channel Selection Problem are updated at a much slower time scale compared to solutions to the Scheduling Problem and the Channel Access Problem. Based on this timescale separation, we first study the solutions to the Scheduling Problem and the Channel Access Problem, given fixed solutions to the Client Association Problem and the Channel Selection Problem. We then study the solutions to the Client Association Problem and the Channel Selection Problem, under the knowledge of how solutions to the Scheduling Problem and the Channel Access Problem react. Thus, solutions to the Client Association Problem and the Channel Selection Problem are indeed joint solutions to all the four problems, and their optimal solutions achieve Pareto-optimal weighted proportional fairness. In addition to solving the four problems, we will show that the solutions naturally turn into distributed algorithms where each client/ AP makes decisions based on local knowledge. V. THE SCHEDULING PROBLEM AND THE CHANNEL ACCESS PROBLEM In this section, we assume that solutions to the Client Association Problem and the Channel Selection Problem, i.e., n(i) and c(n), are fixed.
Since n(i) and c(n) are fixed, values of B i,n(i),c(n(i)) are constant. The optimization problem can be rewritten as s.t. 0 ≤ p n ≤ 1, for all n; φ i,n(i) ≥ 0, for all i; i:n(i)=n φ i,n = 1, for all n, where w n = i:n(i)=n w i , as defined in Section III. We also define z n := m∈M n,c(n) ,c(m)=c(n) w m to be the total weights of clients that are associated with APs that interfere with n, including itself.
This formulation naturally decomposes the optimization problem into two independent parts: maximizing i w i log φ i,n(i) over φ i,n , which is the Scheduling Problem, and maximizing n [w n log p n +(z n −w n ) log(1−p n )] over p n , which is the Channel Access Problem. Thus, we can solve these two problems independently.
We first solve the Scheduling Problem. Theorem 1: Given n(i), c(n), and p n , for all i and n, with the condition i:n(i)=n φ i,n = 1, we have that wi φi,n = wj φj,n , for all i, j such that n(i) = n(j) = n, at the optimal point. By setting φ i,n(i) ≡ w i /w n(i) , the aforementioned criterion is satisfied. The conditions φ i,n(i) ≥ 0, for all i, and i:n(i)=n φ i,n = 1, for all n, are also satisfied. Thus, the Scheduling Problem is solved by setting φ i,n(i) ≡ w i /w n(i) .
We solve the Channel Access Problem next. The following theorem is the direct result of Theorem 1 in [17].
Theorem 2: Given n(i), c(n), and φ i,n , for all i and n, i w i log r i is maximized by setting p n ≡ w n /z n . In summary, when the solutions to the Client Association Problem and the Channel Selection Problem, i.e., n(i) and c(n), are fixed, the AP n should access the channel with probability p n = w n /z n in each time slot and should schedule the transmission for its client i with probability φ i,n = w i /w n whenever it accesses the channel. In addition to achieving the optimal solution to both the Scheduling Problem and the Channel Access Problem, this solution only requires n to know the local information of w m and c(m) for all AP m that may interfere with itself. Thus, this solution can be easily implemented distributedly.
For the distributed scheme, a theorem similar to Theorem 2 shows that the Channel Access Problem is optimally solved by choosing p i = w i /z n(i) .

VI. THE CLIENT ASSOCIATION PROBLEM AND THE CHANNEL SELECTION PROBLEM We now propose a distributed algorithm that solves the Client Association Problem and the Channel Selection
Problem based on the knowledge of optimal solutions to the Scheduling Problem and the Channel Access Problem. These two problems are non-convex and a local optimal solution to the two problems may not be globally optimum, which we will also illustrate by simulations in Section VII. Thus, common techniques for solving convex problems are not suitable for these problems. Instead, the proposed algorithm uses a simulated annealing technique that is based on the Gibbs Sampler [14], which is proven to converge to the global optimum point almost surely. We first give an overview of the technique. We then describe a centralized algorithm that achieves weighted proportional fairness using the Gibbs Sampler. Finally, we discuss how to turn this centralized algorithm into a distributed protocol.
We call a joint solution to both the Client Association Problem and the Channel Selection Problem as a configuration of the system. A configuration is thus fully specified by the AP each client is associated with, and the channel each AP operates in. Define ψ t as the configuration of the system at time t. We define the energy of the system under configuration ψ t , which we denote by U (ψ t ), as the value of i w i log r i when APs and clients choose their channels to operate in and APs to be associated with according to ψ t , and apply the optimal solution to the Scheduling Problem and the Channel Access Problem under ψ t . We then have Finding the joint solution that achieves Pareto-optimal proportional fairness is equivalent to finding the configuration ψ that maximizes U (ψ).
We apply the Gibbs sampler to solve the Client Association Problem and the Channel Selection Problem jointly. At each time t, either a client or an AP is selected according to some arbitrary sequence. The selected client, or AP, then changes the AP it is associated with, or the channel it operates in, randomly, while all other clients and APs make no changes. The solutions to the Scheduling Problem and the Channel Access Problem are then updated according to the new configuration.
We now discuss how the selected client, or AP, changes the AP it is associated with, or the channel it operates in. Let ψ t (n(i) = n) be the configuration where client i is associated with AP n, and the remaining of the system is the same as in configuration ψ t . We can define ψ t (c(n) = c) for AP n similarly. If client i is selected at time t, it changes the AP it is associated with to n with probability e U(ψt(n(i)=n))/T (t) / m e U(ψt(n(i)=m))/T (t) , where T (t) is a positive decreasing function. On the other hand, if AP n is selected at time t, it changes the channel it operates in to c with probability e U(ψt(c(n)=c))/T (t) / d e U(ψt(c(n)=d))/T (t) . [14] proves that this simple randomized approach maximizes U (ψ).
It remains to compute the values of U (ψ t (n(i) = n)) for client i and U (ψ t (c(n) = c)) for AP n. We first discuss how to compute U (ψ t (n(i) = n)). Let w n −i := j:n(j)=n,j =i w j be the total weights of clients, excluding i, associated with AP n. Let z n −i := m∈M n,c(n) ,c(m)=c(n) w m −i . Define which can be thought of as the energy of the system as if the weight of client i were zero. We then define ∆U n i (ψ t ) := U (ψ t (n(i) = n)) − U 0 i . Since in the configuration ψ t (n(i) = n), w m = w m −i for all m = n; , c(m) = c(n), and m = n; and z m = z m −i , otherwise, we have Suppose a client i is selected to change its state at time t, at which time the configuration of the system is ψ t . The probability that i chooses AP n to be associated with is e [U 0 When an AP n is selected by the Gibbs sampler at time t, it changes the channel it operates randomly, with the probability of changing to channel c proportional to e U(ψt(c(n)=c))/T (t) = e [U 0 n (ψt)+∆U c n (ψt)]/T (t) ∝ e ∆U c n (ψt)/T (t) . We note that, to compute ∆U c n (ψ t ), AP n only needs the values of B i,n,c , w i , for each client i that is associated with n, and z m −n , w m for all m ∈ ∪ c M n,c . Thus, ∆U c n (ψ t ) can also be computed using only local information.
Based on the above discussion, it is straightforward to design a distributed protocol (DP) using the Gibbs sampler. DP achieves the Pareto-optimal proportional fairness as t → ∞ almost surely. Further, in DP, all clients and APs in the system only need to exchange information within their local areas, as they only need local information to compute the probability of choosing an AP to be associated with or a channel to operate in. Thus, DP is easily scalable.
In addition to DP, we can also consider a greedy policy (Greedy) that is easier to implement. Greedy works similar to DP, except that when a client i, or an AP n, is selected by the sampler, it chooses the AP that maximizes U (ψ t (n(i) = n)) to be associated with, or the channel that maximizes U (ψ t (c(n) = c)) to operate in, respectively. It is essentially a steepest descent direction approach and is guaranteed to converge to a local optimal point. In addition to simple implementation, Greedy is also consistent with the selfish behavior of clients. Each client i chooses the AP n that maximizes z o +w o , which is indeed the value of r i when i is associated with n. Thus, in Greedy, every client always chooses to associate with the AP that offers the highest throughput.
A similar protocol can also be designed for the distributed scenario where clients, instead of APs, contend for channel access. The protocol also uses the Gibbs sampler as DP. Define z n −i , z m −n , U 0 i (ψ t ), ∆U n i (ψ t ), U 0 n (ψ t ), and ∆U c n (ψ t ) similar as in the server-centric scheme. We can derive that, for the distributed protocol, ∆U n i (ψ t ) =w i log(B i,n,c(n) wi z n −i +wi j:n(j)∈M n,c(n) , c(n(j))=c(n),j =i z n(j)

VII. SIMULATION RESULTS
We have implemented both DP and Greedy algorithms. We also compare their performances against other stateof-the-art solutions. We only present simulation results for the server-centric scheme due to space limitations.
We first introduce the model of channel characteristics in our simulation. We use the ITU path loss model [1] to compute the received signal strength between two devices. If two devices operate in a band with frequency f c and are d meters apart, the received signal strength of a device by the other is proportional to 1 f 2 c d α , where α is the path loss coefficient and is set to be 3.5.
We adopt the simulation settings in [21], which models 802.11b channels, as the base case and compute the characteristics of other channels accordingly. A 802.11b channel operates in the 2.4 GHz band with bandwidth 22 MHz. The bit rate between a client and an AP is 11 Mbps if the distance between them is within 50 meters, 5.5 Mpbs within 80 meters, 2 Mpbs within 120 meters, and 1 Mps within 150 meters. The maximum transmission range of 802.11b channels is hence 150 meters. We assume that two APs interfere with each other if the received signal strength of one AP by the other is above the carrier sense threshold, which, as the settings in ns-2 simulator, is set to be 23.42 times smaller than the received signal strength at a distance of the maximum transmission range. Using the ITU path loss model, two APs interfere with each other if they are within the interference range, which is 150 × (23.42) 1 3.5 = 369 meters for 802.11b channels. For channels other than 802.11b channels, we assume that each of these channels can support four different data rates, corresponding to the four data rates of 802.11b channels. Values of each supported data rate is proportional to the bandwidth of the channel. The transmission ranges of each data rate and the interference range are computed so that the received signal strengths at the boundary of each range is the same to that at the boundary of its counterpart in 802.11b channels. For example, consider a channel that operates in frequency 4 GHz with bandwidth 44 MHz. The bit rate between a client and an AP is 11 × We compare our algorithms, DP and Greedy, against policies that use state-of-the-art techniques for solving the Client Association Problem and the Channel Selection Problem. We compare with [20], which proposes a distributed algorithm for achieving minimum total interference among APs, for the Channel Selection Problem. For the Client Association Problem and the Scheduling Problem, we compare with two techniques. The first technique uses a Wifi-like approach where clients are associated with the closest AP and the AP schedules clients so that the throughput of each client is the same. The protocol that applies both [20] and the Wifi-like approach is called MinInt-Wifi. The other technique is one that is proposed in [21], which, under a fixed solution of the Channel Selection Problem, is a centralized algorithm that aims to find the joint optimal solution to the Client Association Problem and the Scheduling Problem that achieves weighted proportional fairness. This technique first relaxes the Client Association Problem by assuming that each client can be associated with more than one APs and formulates the problem as a convex programming problem. It then rounds up the solution to the convex programming problem and finds a solution to the Client Association Problem where each client is associated with only one AP. For ease of comparison, we use the solutions to the relaxed convex programming problem, which is indeed an upper-bound on the performance of [21]. The protocol that applies both [20] and [21] is called MinInt-PF. The Channel Access Problem is then solved by the optimal solutions based on the resulting solutions of MinInt-Wifi and MinInt-PF, respectively.
In each of the following simulations, we initiate the system by randomly assigning channels to each radio of APs. Each client is initially associated with the closest radio, with ties broken randomly. The system then evolves according to the evaluated policies. We compare the policies on two metrics: the weighted sum of the logarithms of throughput for clients, i∈I w i log r i , and the total weighted throughput i∈I w i r i . All reported data are the average over 20 runs. We first show the simulation results for a simple system that consists of 3 APs and 2 different channels. While this system may be simplistic, it offers insights on the behavior of each policy. We then show the simulation results for a larger system where the list of available channels is gathered from a real-world scenario.
We first consider a system with 3 APs, each with one radio, that are separated by 75 meters and are located at positions (0,0), (75,0), and (150,0). There are 16 clients, all with weights 1.0, and the i th client is located at position (35+5i, 0). We consider two settings for channels: one that with only one 802.11b channel and the other with one 802.11b channel and a channel that operates at frequency 16 GHz with bandwidth 50 MHz. This channel can support higher data rates but has smaller transmission and interference ranges.
Simulation results are shown in Fig. 1. DP and Greedy outperform MinInt-Wifi and MinInt-PF in both evaluated metrics under both 1-channel and 2-channel settings. For the case where there is only one channel, the solutions of the Channel Selection Problem does not have any influence on the results. MinInt-PF does not have good performance because it distribute clients equally to all three APs, which leads to serious contention and collisions within the network. MinInt-Wifi also suffers from the same problem. On the other hand, under DP, all clients are associated with the AP located at (75,0) and therefore contention is avoided. This result suggests that a desirable Fig. 1  For the case where there are two channels, both MinInt-Wifi and MinInt-PF select the APs at (0,0) and (150,0) to operate in the channel at frequency 16 GHz with bandwidth 50 MHz and the AP at (75,0) to operate in the 802.11b channel. This selection is the only one that results in no interference within the network. On the other hand, DP selects the APs at (0,0) and (150,0) to operate in the 802.11b channel and the AP at (75,0) to operate in the other channel. While this selection results in interference between the APs at (0,0) and (150,0), DP actually achieves better performance in both metrics. This is because, in our setting, most clients are gathered around the AP at (75,0) and thus an optimal solution should allow the AP at (75,0) to operate in a channel with higher data rates. This shows that an algorithm that aims to minimize interference among APs may not be optimal because it fails to consider the geographical distribution of clients. Further, although the performance of Greedy is suboptimal, which is because the Client Association Problem and the Channel Selection Problem are nonconvex, it is actually close to that of DP and is much better than those of MinInt-Wifi and MinInt-PF.
Next, we consider a larger system. The system consists of 16 APs that are planned as a 4 by 4 grid. Each AP has 2 radios and adjacent APs are separated by 300 meters. There are 16  We consider the TV white spaces available in New York City [11]. The list of available channels is shown in Table  I. We consider two settings: an unweighted setting where all clients have weights 1.0, and a weighted setting where clients within the region [0, 300]×[0, 900] have weights 1.5 and clients outside this region have weights 0.5.
Simulation results are shown in Fig. 2. For both the unweighted and weighted settings, MinInt-Wifi and MinInt-PF are far from optimum. The total weighted throughputs achieved by the two policies are less than half of those achieved by DP under both settings. The performance of Greedy is close to optimum, whose weighted total throughputs are about 85% and 84% of those by DP for Fig. 2: Performance comparison for a larger system. the unweighted and weighted settings, respectively.

VIII. CONCLUSION
We have studied the problem of achieving weighted proportional fairness in multi-band wireless networks. We have considered a system that consists of several APs and clients operating in a number of available channels, accounting for interference among APs and heterogeneous characteristics of different channels. We have identified that the problem of achieving weighted proportional fairness in such a system involves four important components: client scheduling, channel access, client association, and channel selection. We have proposed a distributed protocol that jointly considers the four components and achieves weighted proportional fairness. We have also derived a greedy policy based on the distributed protocol that is easier to implement. Simulation results have shown that the distributed protocol outperforms state-of-the-art techniques. The total weighted throughputs achieved by the distributed protocol can be twice as large as state-of-the-art techniques. Simulation results have also shown that, while being suboptimal, the performance of the greedy policy is actually close to optimum quite often.