Channel Assignment for Maximum Throughput in Multi-Channel Access Point Networks

Channel Assignment for Maximum Throughput in Multi-Channel Access Point Networks Xiang Luo, Raj Iyengar and Koushik Kar Abstract We consider the uplink channel assignment problem in a multi-channel access point wireless network, with the goal of attaining maximum system throughput. In this setup, a set of orthogonal channels must be assigned to a set of users, where each user splits its power optimally across the channels allocated to it. While the optimal power allocation solution has a water-filling type structure, the optimal channel assignment problem is very challenging due to the non-linear dependence of user throughput on the set of channels assigned to it. Since the optimal channel allocations is computationally intensive to obtain in general, we analyze the system in the two extremal regimes (very high and very low ) and show how the optimal solutions can be obtained in these regimes in a computationally efficient manner. Finally, we demonstrate that the best of the optimal solutions obtained for the two extremes show excellent (close to optimal) performance over the entire range. develop solutions that are computationally efficient, practical, and result in a performance that is close to optimal. The paper is structured as follows. In Section II, we discuss the related work in this space and their relationship with our work. Section III describes the assumptions made in our study of the problem and the resulting optimization problem formulation. In Sections IV and V, we analyze the system throughput in extremal regimes. This analysis motivates the development of algorithms that perform excellently in that they achieve close to the optimal system throughput. Detailed simulation results are presented in Section VI which compare the performance of the proposed algorithms to the optimal solution and intuitive greedy heuristics. I. INTRODUCTION Future generation wireless systems are likely to provide user with simultaneous access to multiple channels. These channels could be a consequence of dynamic spectrum allocation and deallocation [], or PHY layer technologies like OFDMA [3] which decompose a wideband channel into multiple orthogonal narrowband channels. In either case, a multiple channel model is a useful abstraction to study allocation problems in such systems. In this paper, we consider the uplink channel assignment problem for a multi-channel access point system, and develop solutions that maximize the overall system throughput. We view the algorithms developed in this paper as being applicable to networks built around the IEEE 802.6 standard which is widely expected to function over an OFDM/OFDMA physical layer. The dynamic nature of wireless channel qualities and the user-specific differences in perceived channel rates imply that channels may need to be reassigned across users on a frequent basis. However, the optimal channel assignment problem is a challenging problem, due to the complex dependence of the overall system throughput on the channel allocations. For any given channel allocation, a user splits its total power across all channels allocated to it so as to maximize the overall user throughput. The optimum power allocation for a user corresponds to a water-filling type solution over the channels allocated to that user. This results in the user throughputs being complex non-linear functions of the channel allocations. In spite of the complexity and computational difficulty of the optimal channel assignment problem, in this paper we All authors are with the Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 280, USA (email: {kark,luox3,iyengr}@rpi.edu). II. RELATED WORK In [9][0], M. J. Neely et al. consider joint dynamic routing and power allocation problem for a wireless network with time-varying channels and propose several practical control strategies. Their work establishes the capacity region of that the system can stably support and specifies the minimum average power required for network stability among the class of all algorithms. Also, R. L. Cruz et al.[5] study the problem of joint routing, link scheduling and power control to minimize the total average transmission power in wireless multi-hop networks, subject to constraints on the minimum average data rate per link, as well as peak transmission power constraints per node. However, we focus on the problem of channel allocation (on the uplink) to maximize sum throughput across all users in multichannel access point networks. For the OFDMA case, there have been some related work that address the resource allocation problem that is close to our work [][2][3][4][6]. S. Kittipiyakul and T. Javidi et al. [][2] consider the issue of optimal subcarrier allocation in OFDMA, but they assume that the transmit power per subcarrier of each user is pre-determined. In contrast, we consider the joint channel and power allocation question. M. Ergen et al. consider the resource allocation problem for fair scheduling in OFDMA systems [3], however, our contribution lies in the development of channel allocation policy and power allocation that maximizes the whole system throughput. Also, Ian C. Wong et al. [4] addressed the question of how resources should be allocated to users in an OFDMA multichannel system to maximize system throughput. However, the problem that they consider is for the downlink case of the OFDMA system, which is different from our uplink optimal channel assignment problem. D. Kivanc et al. [6] studied the problem of finding an optimal subcarrier and

2 power allocation strategy for downlink communication to multiple users in an orthogonal-frequency-division multiplexingbased wireless system, subject to QoS constraints on the users. In [6], the authors propose heuristics for the problem considered. However, in this paper, we analyze the uplink subcarrier allocation problem in extremal SNR regimes to motivate techniques which achieve near optimal performance. There is a rich body of literature on the subject of throughput-optimal scheduling in wireless networks, for example [4][5][6][7][7][8]. In this paper, we consider the impact of the channel and power allocation across a set of users to maximize sum throughput across all users. The discrete nature of subcarrier allocation coupled with non-linear dependence of throughput on power make the problem considered in this paper more challenging. III. SYSTEM MODEL Our system consists of a set of L users sharing a set of M channels to communicate with an access point (AP). Each user is capable of using multiple channels simultaneously, and the base station (BS) is capable of decoding signals tranmsitted concurrently on multiple channels. However, note that a single channel cannot be used simultaneously by multiple users. Such multi-channel systems are being made possible due ingenious combinations of available hardware and the Software Defined Radio (SDR) paradigm. In the case of OFDM/OFDMA PHYs, the multi-channel operation is implicit. We allow channel conditions to vary across channels as well as users. Channel conditions depend on various factors like fading and interference (from neighboring access point networks), which typically depend on the channel frequency, as well as the user location. Therefore, the attainable rate may differ across channels; moreover, the attainable rate may also depend on the user using the channel. Without loss of generality, we assume that time is slotted and focus on the channel allocation problem across users for a given time slot. It is reasonable to assume that the channel conditions or user population do not change over the duration of a time slot. In the rest of the paper, we will refer to as the noise power seen by user i on channel j at the specific decision instant; note that can include intrinsic receiver noise as well as interference noise from neighboring channels and networks. Note that the noise powers are typically functions of time, since fading and interference levels at any location vary with time. These variations are expected to be more pronounced when the users are mobile. Let P i denote the aggregate transmission signal power (over all channels used by the user) corresponding to user i. This signal power is split across different channels that user i uses. A. Problem Formulation Recall that multiple channels can be assigned to a single user, but a single channel cannot be shared by multiple users. Therefore, a valid assignment of channels to users corresponds to a one-to-many mapping from users to channels. In this paper, we refer to such an assignment as a poly-matching in the user-channel bipartite graph (see Figure ). Let Φ denote User User 2 User 3 User 4 Channel Channel 2 Channel 3 Fig.. A poly-matching: The figure shows one valid poly-matching for 4 users and 3 channels. (Note that the poly-matching is represented by the bold edges.) the set of all poly-matchings in the user-channel graph. The throughput of user i on a channel j assigned to it is of the form B j log( + κ pij ), where B j and κ are constants. For ease of exposition, we will assume B j = B j, and κ =, although the analysis and algorithms that we present in this paper can easily be extended to the more general case. The throughput maximization problem for the entire system can be posed as: max L subject to: i= j:(i,j) φ log( + p ij ) () j φ i p ij P i i, (2) p ij 0 i, j, (3) φ Φ. (4) Note that for a given poly-matching φ, the above problem reduces to the optimal power allocation problem for each user, whose solution corresponds to a water-filling across the different channels assigned to the user. The problem posed above is much more complex, however, since it corresponds to a joint channel and power allocation problem: it requires us to find the channel assignment (poly-matching) that will yield the best system throughput under optimal power allocations for that channel assignment. A naive approach to solve this problem would be to enumerate all poly-matchings, compute the attainable throughput for a poly-matching by running the water-filling algorithm, and then pick the poly-matching that yields the maximum throughput value. However, since the number of poly-matchings is in general exponential in the size of the user-channel bipartite graph, this naive approach is computationally very expensive, and not feasible for large number of channels or users. Thus our goal is to obtain optimal or near-optimal channel assignments in a computationally efficient manner. B. Solution Approach The complexity of the channel assignment problem arises due to its close coupling with the optimum power allocation problem. This results in a non-linear dependence of the system throughput on the channel assignments. In other words, the throughput gain for a user on being allocated an additional channel depends strongly on the channels that have already been assigned to the user, and this dependency is quite

3 complex in general. In view of this, our goal is to develop approximate solutions that can be computed efficiently, but yet results in near-optimal throughput. Towards this end, we will analyze the throughput maximization question posed above in the two extremal regimes. We argue that in both the high and the low regimes, the optimal channel assignment (poly-matching) can be obtained efficiently using appropriate matching algorithms. We then show, through simulation experiments, the better of the two algorithms developed for the two extreme points performs well over the entire range of. User (M sub nodes) α ijk (Edge Weights) Channel Channel 2 IV. THROUGHPUT ANALYSIS IN THE HIGH REGIME In section, we analyze the throughput attained by a user i in the high regime to motivate an algorithm that computes a channel assignment (poly-matching) that is optimal in this regime. Consider any user i, and let φ i = {j : (i, j) φ} denote the set of channels assigned to user i. Let k i = φ i. In the high regime, P i j φ i. This implies that if power allocation is done to optimize user throughput, the user will use all the channels assigned to it, and the power allocations will correspond to a water-filling solution, as characterized by p ij + = λ i j φ i. (5) Summing over all the k i channels, we obtain λ i = P i + N i k i, (6) where P i is the aggregate transmission power of user i, and N i, the aggregate noise power of user i, is defined as N i = j φ i. If we represent the throughput attained by user i as U i, then we can write U i = j φ i log( + p ij ) (7) = j φ i log( p ij + ) (8) = j φ i log( P i + N i k i ) (9) j φ i log( P i k i ), (0) where the approximation comes from the fact that in the high regime, P i N i. Hence we can write the throughput for user i as U i = k i log(p i ) k i log(k i ) j φ i log( ). () The importance of writing this expression for the throughput is that it allows us to quantify the incremental utility of allocating the jth channel to user i. In general, this incremental utility is a function of i, j, k i. Specifically, consider the incremental utility of allocating channel j to user i, when k channels have already been allocated to it. Then, using the throughput expression in (), the incremental utility in this case, denoted by α ijk, is expressed as follows (note that in both cases Fig. 2. User L (M sub nodes) The constructed graph G. Channel M before as well as after the allocation of channel j the total power P i available at user i is divided using water-filling technique across the set of allocated channels): α ijk = log(p i ) (k log(k) (k )log(k )) log( ). (2) In the above expression, (k )log(k ) at k = should be interpreted as 0. We note that the expression for incremental utility expression includes only terms specific to the added channel j (log( )), the number of channels already allocated (k), and the total power of user i, P i. Thus the incremental utility expression does not depend on the exact set of channels allocated to user i, but only on the size of that set (k). This allows us to set up the following graph formulation of the throughput maximization problem in the high regime. In Figure 2, the L nodes representing the users are split up into M sub-nodes, one for each of the channels. The channels are represented separately using M nodes, as usual. All possible edges between the user sub-nodes and channels are drawn, with edge weights computed using (2). Given this construction, note that a matching in the constructed bipartite graph G corresponds to a poly-matching in the original graph. Further, it can be verified that the edge-weights exhibit a decreasing property in k, i.e., α ijk > α ij(k+) for any k. If (i, j, k) denotes the edge between the kth sub-node of user i and the channel j, then the decreasing property of the edgeweights imply that a maximum weight matching [2] (with α ijk as the edge-weights) in G will prefer edges that correspond to a lower k, for the same i and j. Thus in a maximum weight matching, for any user i, there will be a k i such that sub-nodes,..., k i, will be matched, and sub-nodes k i +,..., M, would not be matched. This line of argument can be extended further

4 to show that a maximum weight matching in G corresponds to the poly-matching that maximizes the sum of user throughputs, where the user throughputs are defined by (). Therefore, in the high regime, the optimum channel assignment can be calculated by computing the maximum weight matching in the constructed bipartite graph G, the complexity of which is O(L 3 M 3 ) using the classical Hungarian algorithm [8]. The details of the algorithm, which we call the High- -Optimal (HSO) Algorithm, are specified in Algorithm. Algorithm : The High--Optimal (HSO) Algorithm Step : Compute edge-weights α ijk, i, j, k, as follows: If k =, then α ijk = log(p i ) log( ); Else α ijk = log(p i ) (k log(k) (k )log(k )) log( ); Step 2: Use maximum weight matching on the bipartite graph G with α ijk as the edge-weights, to obtain the optimal channel assignment. Step 3: For each user, obtain the optimal power allocations over the channels assigned to it using water-filling. V. THROUGHPUT ANALYSIS IN THE LOW REGIME In the low regime, we approximate the objective function as log( + p ij ) p ij, (3) using the approximation log( + x) x when 0 < x. Further, if we assume that all values are distinct, then for small enough, each user will allocate all its power in a single channel the one with the smallest among all channels assigned to the user. More precisely, if the values differ at least by ǫ, then for P i < ǫ, user i will allocate all of its power to the minimum-noise channel it is allocated, for maximum throughput. (Note that this situation is the opposite of the high case, where the user will typically use all channels assigned to the user.) Using this fact, and (3), we see that the channel assignment policy for maximum throughput in the low regime corresponds to a maximum weight matching in the complete bipartite graph of users and channels, with edge-weights β ij = Pi. This maximum weight matching can be computed in O({max(L, M)} 3 ) time, using the Hungarian algorithm [8]. Note that if the number of channels is more than the number of users, the matching algorithm will leave a number of channels unassigned to any user. In practice, P i can be larger than the minimum difference in the noise-levels, and thus leaving available channels unassigned can led to considerable wastage of resources. Therefore, in the algorithm described below as Low--Optimal (LSO) Algorithm, we run the matching iteratively, leaving out all channels assigned in previous iterations, until all channels are allocated. Algorithm 2: The Low--Optimal (LSO) Algorithm Step : Compute edge-weights β ij, i, j, as β ij = Pi. Step 2: Until all channels are assigned, do the following: ) Compute the maximum weight bipartite matching among the users and unassigned channels, with β ij as the edge-weights. 2) Assign channels to users based on the matching, and eliminate them from further consideration. Step 3: For each user, obtain the optimal power allocations over the channels assigned to it using water-filling. VI. PERFORMANCE ANALYSIS AND DICUSSION In this section, we evaluate the performance of our channel assignment algorithms in multichannel access point networks through simulations. More specifically, from simulation results, we demonstrate that the HSO and LSO algorithms actually achieve the optimal channel assignments in high regime and low regime, respectively. We also study the performance of two simple intuitive heuristics (to be described shortly) and compare them with the HSO and LSO algorithms. In the first heuristic, which we call the Incremental Max- Throughput (IMT) Heuristic, we assign channels (to users) one by one, in any order, with the user chosen such that the assignment yields the maximum additional throughput across all users. The additional throughput gained by assigning a channel j to user i can be computed simply as follows. We first compute the current throughput by running the water-filling power allocation solution for the current channel assignment. We then tentatively add channel j to user i, and re-run the water-filling algorithm; comparing this throughput with the current throughput provides us the gain of assigning channel j to user i. We then assign channel j to the user which results in the maximum gain. In the second intuitive heuristic, which we call the Incremental -Balancing (ISB) Heuristic, we again assign channels (to users) one by one, in any order, with the user chosen such that the ratio of the total power and the total noise is balanced across all users, as much as possible. To determine which user a channel j should be assigned to, we do the following. For every user i, we tentatively assign channel j to user i, and then compute the maximum difference in the (P i /N i ) ratio over all channels i. We then assign channel j to the user such that this difference is minimized. Figures 3-6 show the simulation results for 4 different system models, which consist of 3 users and 6 channels, 3 users and 9 channels, 4 users and 8 channels, 5 users and 0 channels, respectively. In all simulations, we choose nij from Gaussian distribution N(0, σ 2 ); thus we have E( ) = σ 2. For each user i, the maximum power P i is chosen from the Uniform distribution U(,.5); thus E(P i ) =. In the simulations, σ 2 is changed (keeping P i fixed) to generate a wide range of environments. For the same value of σ 2, simulations are run several times; the performance numbers shown in the figures across different

5 Performance Ratio Performance Ratio 0 2 0 0 0 0 0 2 0 2 0 0 0 0 0 2 Fig. 3. Simulation Results for 3 Users and 6 Channels Fig. 4. Simulation Results for 3 Users and 9 Channels values correspond to the average performance for that. In the figures, the x-axis corresponds to the, plotted in a semi-log scale. The y-axis corresponds to the ratio of the average throughput attained by an algorithm/heuristic and the maximum throughput attainable (solved by complete enumeration over all possible poly-matchings). Note that the curves for Max(HSO, LSO) in the figures shows the best performance amongst the HSO and LSO algorithms. From the figures, we see that the HSO algorithm achieves the optimal channel assignment (performance ratio is ) under high. In fact, the performance ratio of HSO it almost optimal when is close to unity or higher. The figures also show that the LSO algorithm performs near-optimally when is low; its performance worsens as increases, as expected. We observe that the IMT heuristic performs fairly well at high, but its performance is poor at low values of. The ISB heuristic performs poorly at both high and low, although its performance improves steadily as increases. Note that out of the two greedy algorithms, the IMT heuristic is computationally quite complex, since it requires us to solve the water-filling algorithm O(LM) times. The ISB heuristic is computationally less complex, but with worse performance. We observe that the better of the HSO and LSO algorithms performs optimally over the entire range of considered. This performance is also considerably better than that of the incremental heuristics. In practice, therefore, we can run both the HSO and LSO algorithms, and pick the better solution; this would result in near optimal performance no matter what the value is, at only a small computation cost. REFERENCES [] T. Kamakaris, M. Buddhikot and R. Iyer, A Case for Co-ordinated Dynamic Spectrum Access In Cellular Networks, Proceedings of First International Conference on Dynamic Spectrum Access Networks, DyS- PAN 2005, Nov. 2005, Baltimore, USA. [2] T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms, McGraw-Hill, 990. Performace Ratio Fig. 5. 0 2 0 0 0 0 0 2 Simulation Results for 4 Users and 8 Channels [3] Draft IEEE Standard for Local and Metropolitan Area Networks, Part 6 Air Interface for Fixed Broadband Wireless Access Systems. IEEE Std 802.6-2004. [4] A. Eryilmaz and R. Srikant, Fair Resource Allocation in Wireless Networks using Queue-length based scheduling and Congestion Control, Proceedings of INFOCOM 2005, Miami, March 2005. [5] L. Tassiulas, A. Ephremides, Stability properties of constrained queueing systems and scheduling for maximum throughput in multihop radio networks, IEEE Transactions on Automatic Control, Vol. 37, No. 2, pp. 936-949, December 992. [6] L. Tassiulas, P. Bhattacharya, Allocation of interdependent resources for maximum throughput, Stochastic Models, 999. [7] N. McKeown, V. Anantharam and J. Walrand, Achieving 00% Throughput in an Input-Queued Switch, Proceedings of IEEE Infocom 96, San Francisco, Mar 996. [8] H. W. Kuhn, The Hungarian Method for the assignment problem, Naval Research Logistic Quarterly, 2:83-97, 955. [9] M. J. Neely, Energy Optimal Control for Time Varying Wireless Networks, IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 295-2934, July 2006. [0] M. J. Neely, E. Modiano, and C. E. Rohrs, Dynamic Power Allocation and Routing for Time Varying Wireless Networks, IEEE Journal on Selected Areas in Communications, Special Issue on Wireless Ad-Hoc Networks, vol. 23, no., pp. 89-03, Jan. 2005. [] S. Kittipiyakul and T. Javidi, Subcarrier allocation in OFDMA systems:

6 Performance Ratio 0 2 0 0 0 0 0 2 Fig. 6. Simulation Results for 5 Users and 0 Channels beyond water-filling, Asilomar Conference on Signals, Systems, and Computers, November 2004. [2] S. Kittipiyakul and T. Javidi, A Fresh Look at Optimal Subcarrier Allocation in OFDMA Systems, IEEE Conference on Decision and Control (CDC 2004), Dec 2004. [3] Mustafa Ergen, Sinem Coleri, and Pravin Varaiya, QoS Aware Adaptive Resource Allocation Techniques for Fair Scheduling in OFDMA Based Broadband Wireless Access Systems, IEEE Trans on Broadcasting 94(4): 362-370, Dec. 2003. [4] Ian C. Wong, Zukang Shen, Brian L. Evans, and Jeffrey G. Andrews, A Low Complexity Algorithm for Proportional Resource Allocation in OFDMA Systems, IEEE Workshop on Signal Processing Systems (SIPS04), October 3-5, 2004. [5] R. Cruz and A. Santhanam. Optimal routing, link scheduling, and power control in multi-hop wireless networks. IEEE Proceedings of INFOCOM, April 2003. [6] D. Kivanc, Guoqing Li, and Hui Liu, Computationally efficient bandwidth allocation and power control for an OFDMA system, IEEE Journal on Selected Areas in Communications, November 2003. [7] K. Ross and N. Bambos, Local Search Scheduling Algorithms for Maximal Throughput in Packet Switches Optimizing, Proceedings of IEEE INFOCOM 2004, Hong Kong, China, March 2004. [8] K. Ross and N. Bambos, Projective Cone Schedules in Queueing Structures: Geometry of Packet Scheduling in Communication Network Switches, Proceedings of Allerton Conference on Communication, Control and Computing, Allerton Park, Monticello, IL, October, 2002.