Optimal Channel Width Adaptation, Logical Topology Design, and Routing in Wireless Mesh Networks
© L. Li and C. Zhang 2009
Received: 23 December 2008
Accepted: 16 March 2009
Published: 4 May 2009
Radio frequency spectrum is a finite and scarce resource. How to efficiently use the spectrum resource is one of the fundamental issues for multi-radio multi-channel wireless mesh networks. However, past research efforts that attempt to exploit multiple channels always assume channels of fixed predetermined width, which prohibits the further effective use of the spectrum resource. In this paper, we address how to optimally adapt channel width to more efficiently utilize the spectrum in IEEE802.11-based multi-radio multi-channel mesh networks. We mathematically formulate the channel width adaptation, logical topology design, and routing as a joint mixed 0-1 integer linear optimization problem, and we also propose our heuristic assignment algorithm. Simulation results show that our method can significantly improve spectrum use efficiency and network performance.
Wireless mesh networks (WMNs) consist of a multihop backbone of mesh routers which collect and relay the traffic generated by mesh clients . A fundamental obstacle to building large-scale multihop wireless networks is the insufficient network capacity when route lengths and network density increase due to the limited spectrum shared in the neighborhood . The use of multiple radios which tuned into different channels can significantly improve the network capacity by employing concurrent transmissions under different channels, and that motivates the development of new protocols for multi-radio multi-channel (MR-MC) mesh networks.
Each of the nodes from 1 to 9 is assumed to generate a flow of same throughput towards the gateway, node 10. Intermediate nodes act as traffic generators as well as traffic routers at the same time. So different links carry different traffic loads. In Figure 2(a), the number above each link indicates the expected load on the link. For example, link (5, 6) has a load of since it forwards flows originating from nodes 1 to 4 and the flow generated by node 5 itself. Obviously, the bottleneck collision domain consists of links (6, 7), (7, 8), (8, 9), and (9, 10), and hence limits the throughput for each flow.
We assume the total available spectrum is 60 MHz wide, and each 1 MHz spectrum can deliver 1 Mbps data rate. Here we consider static spectrum assignment scheme, that is, channels are assigned to interfaces/links on a long-term basis. In Figure 2(b), we first investigate the case that the whole available spectrum is divided into three 20 MHz-wide non-overlapped channels. So at least two links among (6, 7), (7, 8), (8, 9), and (9, 10) will be assigned to the same channel. As Figure 2(b) shows, the optimal scheme is to assign a same channel to link (6, 7) and (7, 8), and assign the other two channels to (8, 9) and (9, 10), respectively. Under this scheme, links (6, 7) and (7, 8) become the bottleneck and every flow can obtain the throughput up to 20/13 . In Figure 2(c), we then investigate another case that four 15 MHz-wide channels are available. Now no two links will interfere with each other. Obviously, the bottleneck link is (9, 10), and every flow can get the throughput up to 5/3 Mbps, which is better than the previous case.
Note that flows could not benefit from the enhanced capacity without first reducing the bottleneck wireless links. By optimally adjusting channel width for every link, we can get the most efficient spectrum assignment scheme as Figure 2(d) shows. The spectrum that every link uses exactly matches its traffic load. Now the throughput for every flow can get up to 2 Mbps. Compared with the previous two fixed-width assignment schemes, channel width adaptation can improve the network performance by 30% and 20%, respectively.
Motivated by the above example, we strongly advocate the channel width adaptable network architecture. Briefly speaking, the advantages of channel width adaptation are two-fold. On one hand, we can distribute the traffic as evenly as possible across the spectrum in a fine granularity to achieve channel load balance. On the other hand, in a scenario with many interfering links, by "creating" more small-width orthogonal channels, we can greatly reduce the phenomena of contention and collision, and therefore improve throughput as a result of fewer back-offs and reduced interference. Another motivation for the channel width adaptable network architecture is the recent open spectrum effort  made by the spectrum regulation authority such as FCC. Because of the variable widths of "white space" unoccupied by licensed users, we believe channel width adaptation will become one of the most important functions for cognitive radio networks in future open spectrum environment.
The characteristic of wireless mesh networks  makes it attractive and feasible to use channel width adaptation. First, in WMN, each mesh router aggregates traffic flows for a large number of mobile clients, and therefore the aggregate traffic load changes infrequently, which offers the predictability for assigning channel width in term of traffic pattern and permits capacity optimization based on estimated traffic demand. Second, mesh nodes (or routers) are usually static and have no power constraints, and therefore physical topology changes only occur due to occasional node failures, or addition of new nodes. Thus channel width adaptation can be implemented on a long-term basis without requiring resynchronization of interfaces for every packet. Third, some mesh routers are used as gateways to connect the wired network, and most traffic is between the mesh clients and the wired networks through these gateways. So the traffic distribution in WMN is typically skewed as the example in Figure 2 shows: gateway nodes would form the bottlenecks since more and more flows contend for the bandwidth as they are forwarded closer to gateways. Channel width adaptation will surely promise great flexibility to accommodate such skewed traffic distribution.
In this paper, we address how to optimally adapt channel width in IEEE802.11-based multi-radio multi-channel wireless mesh networks. We mathematically formulate the channel width adaptation, logical topology design, and routing as a joint optimization problem. Our mathematical formulation not only takes into account the issues in traditional MR-MC mesh networks, such as the number of available interfaces, the interference constraints, and the expected traffic load, but also determines at what center frequency and how wide a spectrum band an interface should use. Extensive simulations show that channel width adaptation can significantly improve spectrum use efficiency and network performance.
The rest of the paper is organized as follows. Section 2 reviews the related work. Section 3 presents the network model and Section 4 formulates the problem as a mixed integer nonlinear programming. In Section 5, we convert the problem into an equivalent mixed 0–1 integer linear programming and propose a suboptimal heuristic solution. Simulation results are presented in Section 6, and Section 7 concludes this paper.
2. Related Work
There exists a wide range of related works aiming to design efficient channel assignment algorithms for multi-radio multi-channel mesh networks.
Raniwala proposed a static centralized channel assignment algorithm in , and in , an improved distributed channel assignment algorithm with load-balance routing was proposed. In , channels are allocated so as to minimize the maximum number of interfering links within each neighborhood, subject to the constraint that the logical topology graph should be K-connected. In , Kyasanur and Vaidya proposed a hybrid channel assignment strategy, easing the channel synchronization. Literature  proposed a routing protocol which incorporates a routing metric taking account of both the loss rate and the channel diversity of links along the path. All the above algorithms are based on heuristic methods, not mathematical formulations.
Many other works formulate the problem as a joint mathematical programming. In , Alicherry et al. formulated a joint channel assignment and routing problem for the MR-MC network, with the aim of maximizing network throughput subject to the proportional fairness constraints. Literature  provided necessary conditions of the feasibility of rate vectors and used a fast primal-dual algorithm to derive upper bounds of the achievable throughput. In , two models that maximize the number of logical links that can be active simultaneously were proposed, subject to interference constraints. In , the MR-MC mesh architecture called TiMesh was proposed, which formulates the logical topology control and interface assignment as a joint optimization problem. All the above works assume channels of fixed predetermined width.
Literature  proposed a spectrum sharing model for cognitive radio networks based on mixed integer nonlinear programming with the objective of minimizing the required network-wide spectrum resource for a set of user sessions, and developed a near-optimal algorithm based on the sequential fixing procedure. It was mentioned in  that equal band division of the spectrum yields suboptimal performance and thus it calculated an optimal global band partition. The significant difference between  and ours is that  only tries to obtain a global spectrum regulation for the whole networks so that all nodes can use only one spectrum partition style, while in our architecture we can adjust channel width flexibly across nodes (i.e., different nodes may use different spectrum partition styles), which offers further flexibility.
Literature  first systematically studied the issues of channel width adaptation. Using commodity 802.11 hardware, it gave a method to generate signals of different channel widths by changing the frequency of the reference clock that drives the frequency synthesizer of the radio front end circuitry, which can be configured dynamically purely in software. And through detailed measurements in controlled environments, it then preliminarily identified several benefits of channel width adaptation in many metrics of wireless networks: range and connectivity, power consumption, network capacity and fairness. Finally, it proposed a channel width adaptation algorithm, called SampleWidth, for two communicating nodes. In , three centralized channel width adaptation algorithms using ILP, LP-based packing and greedy raising were proposed for WLAN to improve network capacity and per-client fairness. Literature  designed a dynamic channel width allocation protocol called b-SMART for cognitive radio networks. Using the concept of time-spectrum block, the spectrum allocation is reduced into the problem of packing time-spectrum blocks into a two-dimensional time-frequency space. The algorithm of  resided in the MAC layer and required advanced radio hardware with fast switching and channel width adaptation ability on a packet-by-packet basis, significantly increasing the signaling overhead due to the fast coordination. In our architecture, channel width adaptation is on a long-term basis (e.g., every several minutes or hours), hence does not require resynchronization of interfaces for every packet and the modification of IEEE802.11 MAC protocols, and thus becomes more practical for current available commercial hardware and easy to be used in wireless backbone mesh networks.
3. Network Model and Problem Formulation
We model the wireless mesh networks by an undirected graph , where denotes the set of all vertices and denotes the set of all edges. Each vertex represents a wireless mesh node equipped with network interface cards, and we use to denote the th interface of node , where . For any two nodes , if node is within the communication range of node , then there is a physical link between and . We assume that all links are bidirectional.
Note that every node has multiple interfaces which can be tuned into different portions of the spectrum, so there may exist zero, one, or more logical links between two neighboring nodes. Then based on the graph , we develop another radio-based graph , where and . We call the links in physical links and the links in logical links. The logical link will exist in the final logical topology after spectrum allocation if and only if the th interface of node and the th interface of node operate on the same portion of spectrum.
We assume that each interface can only be tuned into a contiguous segment of the available spectrum. Due to the hardware constraint, the possible channel widths are some discrete values in the range of . So it is reasonable to partition the whole available spectrum into a series of sequential small-width non-overlapped spectrum blocks. We denote the set of blocks as and the size of a spectrum block as . So the problem of channel width adaptation is equivalent to the contiguous spectrum blocks allocation. For example, in Figure 2, we can set , and the whole available 60 MHz-wide spectrum will be divided into 30 blocks. Link will be assigned the block 22 to block 30 and link will be assigned the block 14 to block 21 in the scheme of Figure 2(d). According to Shannon's capacity theorem , we also reasonably assume that the achievable data rate is proportional to the assigned channel width, that is, the number of spectrum blocks allocated, and we let be the link-layer data rate that one spectrum block can deliver.
We use to denote the set of physical links that are in the interference range of link . Note link also indicates . We assume that the non-overlapped spectrum bands are orthogonal, that is, simultaneous use of non-overlapped spectrum blocks in the same area will not interfere. Though there may exist adjacent channel interference due to improper signal processing at the wireless cards and poor filter characteristics, we believe with the advance of radio technology, adjacent channel interference can be avoided to a large extent, and even partially overlapped channels with variable width can be further exploited in the future.
We assume that a reasonable statistical traffic demand matrix is available. And let denote the traffic demand between the source and destination pair , where . Our aim is to design schemes to maximize the capacity of the network. The network capacity cannot be simply measured by the total throughput of all traffic flows. Optimizing such metric may lead to starvation of some flows which originate far from gateways. We therefore need to consider some fairness constraints. Similar to , we adopt the proportional fairness, that is, the same portion of traffic demand will be satisfied for every flow . So we want to find the schemes that traffic of every flow can be routed for the largest possible . Other kinds of fairness constraint like the lexicographical max-min fairness  can also be adopted.
spectrum block assignment: how to efficiently assign contiguous spectrum blocks to each interface?
routing: how to optimally route the traffic to achieve load balance across different links?
4. Joint Topology Design, Spectrum Assignment, and Routing
In this section, we describe how we formulate the logical topology design, contiguous spectrum block assignment, and routing as a joint optimization problem. We will use the letter like to denote a vector, and use to denote the th element of the vector .
4.1. Contiguous Spectrum Block Allocation
where is the th element of . For example, in Figure 2(d), assuming node 9 uses its nd interface to communicate with node 10, we have while the other elements are equal to zero.
It is possible some radio interfaces do not take part in any communication, so in this case, in constraint (4), and can be zero. Constraint (5) means that the lower end of the spectrum segment should locate lower than the upper end. And in constraint (6), without loss of generality, we further assume that the spectrum segment that interface uses locates lower than that of . Now using and , we can redefine as follows.
Using the constraints (3) to (8), we can fully characterize the contiguous spectrum block allocation. Note we can treat as continuous real vectors since we can infer to be binary vectors from the above constraints.
4.2. Logical Topology Formulation
Vectors and (thus ) can fully characterize the logical topology formulation. The link will exist in final logical topology only when the interfaces and operate on the same set of spectrum blocks. Then we use variable to denote whether the logical link will exist, that is,
where is the exclusive OR (XOR) operator. It is easy to verify the above correspondence. If there is some spectrum block that interface uses while does not or uses while does not, that is, , constraint (11) will imply that . Otherwise, for , constraint (10) will imply that . Note we can also treat as continuous variables.
In multihop WMNs, a source node may need a number of relay nodes to route the data traffic towards its destination node. We need to compute a network flow that associates with each logical link valued , where denotes the traffic data rate for the source and destination pair that is being routed via the logical link in the direction from to , assuring the times of the traffic load valued for every source and destination pair can be routed.
which means if node is the source of the flow, the net flow sent by node should be equal to . If node is the destination of the flow, it should be equal to . For the intermediate relay node, the net flow should be 0. Note a feasible network flow also guarantees that the final logical topology is connected.
The above constraint is only valid for the multi-path routing, which can take advantage of load balancing. We also investigate the single-path routing, which needs more constraints besides (13). We define a binary routing variable for all and for all . The variable will be equal to 1 if the flow from source to destination is only routed via the logical link in the direction from to ; otherwise it will be equal to 0. So should satisfy
4.4. Interference Issues
4.5. Capacity Constraints
The fixed amount of spectrum provides limited capacity that will be shared among the links in interference range. First, we define a real variable as the link utilization for every logical links , that is, the fraction in one unit time that link is active. Remember that we assume channel capacity is proportional to the number of spectrum blocks it used. So should satisfy the following constraints:
The term on right-hand side of constraint (20) is the total traffic rate from all source and destination pairs that is routed over link , which is equal to the link utilization multiplies the channel capacity . Since can be 0 (when the logical link does not exist in the final logical topology, that is, ), we use constraint (21) to set to be 0 in that case.
Extending the sufficient condition for the existence of inference-free schedule of , we have, for any ,
4.6. Objective Function
Now given the topology graph , the parameters , and for all source and destination pairs in , we can state our problem formally using (3)-(23). However, note that many terms such as in (7), in (10) and (11), and in (20) are nonlinear. Even relaxing the binary constraints of (3) and (14), the problem is still nonconvex. So the above programming is a mixed-integer nonconvex program and generally it is not easy to be solved.
5. Solving the Problem
In this section, we first use some linearization techniques to convert the original mixed-integer nonlinear programming into a mixed-integer linear programming. Then we show how to choose the optimal solution with least interference. Finally we propose our heuristic MILP-based iterative local search algorithms.
5.1. Equivalent 0–1 Mixed-Integer Linear Programming
The validity of the above methods can be easily verified by enumerating all possible combinations of and . We take as the example, where and are two binary variables. When , the first linear constraint will imply , and the third linear constraint will imply , so we can get . When , or , the first/second constraints will imply , and the third and the fourth constraints will imply , so . Finally when , the first and the second constraint will imply , and the fourth constraint will imply , and we can conclude that . So the four linear constraints are exactly equivalent to the original nonlinear constraint. And note we can treat as real variables. The other two methods can be verified in the similar way.
In the original programming of Section 4, , , and are explicitly declared binary vectors, while , , and can be directly or intermediately implied to be binary vectors or binary variables from and . is a non-negative real variable with an upper bound valued 1, and is also a non-negative real variable upper bounded by . So it is possible for us to convert all the nonlinear terms into linear ones. For example, for the nonlinear term in (10) and (11), we can first introduce auxiliary variables for all , and then replace the constraint (10) and (11) with the linear constraints as follows:
By applying the above three methods to convert all nonlinear constraints into linear ones, we will get a mixed 0-1 integer linear programming (which is called as MILP-1). The programming MILP-1 has binary integer variables if we use multipath routing and additional binary integer variables if we use single path routing. We can use the traditional branch-and-bound algorithms  or use commercial software solver such as LINDO  and CPLEX  to solve the problem.
5.2. The Optimal Scheme with Least Interference
The above example suggests that we should select a solution that can minimize interference from all solutions which may be produced by MILP-1, that is, all solutions attaining the same optimal valued of . First we adopt following weighted metric to quantify the total interference.
Note that the metric in (25) is nonlinear, but we can easily linearize it via the techniques in Section 5.1 since is an implied binary variable and is a nonnegative real variable with upper bound . Thus the new programming is still a mixed integer linear programming. We call the modified programming MILP-2.
5.3. Heuristic MILP-based Iterative Local Search Algorithm
It is well known that the computational complexity of a mixed integer linear programming mainly depends on the number of integer variables . So for large-scale networks, it will not be trivial to find the optimal solutions to MILP-1 and MILP-2. So we need to make some tradeoff between the performance improvement and computation complexity. In this section, we present our heuristic suboptimal algorithm.
Our heuristic algorithm is an iterative local search algorithms  in which the basic idea is to start with an initial feasible solution and then make modifications to improve its quality using the original MILP. In this section, we only assume that the multipath routing is used, and all nodes are equipped with same interfaces.
We initially partition the whole available spectrum into segments with approximately same size. Then we will assign the first spectrum blocks of each segment to the interfaces of every node. For example, if we have 30 spectrum blocks, and , we will assign blocks 1-6, blocks 11–16, and blocks 21–26 to the first, second and third interface of every node, respectively. Obviously, the network is full connected and only the logic links in the set are preserved.
Then we run the programming MILP-1 on the full connected networks under the given initial spectrum assignment to obtain an initial load balance routing. Note here that MILP-1 becomes a linear programming. With the initial spectrum assignment and routing, we will iterate to create a sequence of solutions in an attempt to gradually improve the network performance.
We should adopt some randomness to escape from the local optimum. So then we randomly choose a logical link from the most congested links and try to adjust the spectrum allocation of all interfaces in the interference range of nodes and . The adjustment is conducted by running a modified version of MILP-1 and MILP-2, where the variables are only a subset of variables of the original problem, while the values of others are kept as constant as those in the previous iteration. Note only that the variables , , and are what we concern about while others are only intermediate variables. For any radio interface where , we mark , as variables of the new iteration. We also mark for all , for all to be variables. The modified problem has much fewer integer variables than the original one, so we can solve it easily by branch-and-bound algorithm. It can be viewed as the local search process.
6. Performance Evaluation
In this section, we compare the performance of our proposed channel width adaptable network architecture with the traditional multi-radio multi-channel networks using fixed-width channels. We also discuss the impact of some system parameters on the network performance.
The simulation is conducted by NS-2 simulator . We use the methods described in  to add multi-interface support and extend the channel module to enable channel width adaptation. The following are the default settings for simulation. We use IEEE802.11 DCF as the MAC layer, and RTS/CTS mechanism is enabled. The two-ray propagation model is used to model the path loss. The transmission range is set to be 250 m, and the interference range is 550 m. The total available spectrum is assumed to be 120 MHz- wide, and each node is equipped with three interfaces. For our channel width adaptable architecture, we set the default spectrum block size to be 5 MHz, and set and to be 5 MHz and 50 MHz respectively. The default routing scheme is multi-path routing. In our implementation of the multipath routing in NS-2, every node forwards data packets across different links with the probability proportional to the routing flows calculated by our programming.
6.1. Optimal and Suboptimal Solutions on Grid Topology
We first present the results obtained by the optimal branch-and-cut solver  and our heuristic MILP-based iterative local search algorithm on the grid topology. We also investigate the performance of MR-MC networks using fixed-width channels, whose solution can be obtained from our MILP programming by adding the constraint . We repeat our simulation on the grid topology for 10 randomly generated traffic profiles. In each profile, we randomly chose twelve source and destination node pairs to generate UDP (User Datagram Protocol) sessions. Each has the transmission demand uniformly distributed between 1 Mbps and 5 Mbps. Then we change every flow's rate proportionally until the network can satisfy 90% of the injected traffic. The metric we examine is the total useful throughput across all sessions.
6.2. Comparison with "Hyacinth" Architecture
"Hyacinth" is a typical MR-MC mesh networks. A static centralized fixed-width channel assignment algorithm for "Hyacinth" architecture is proposed in . With the assumption that most traffic is between the mesh clients and the gateway nodes, it first estimates the total expected load on each virtual link by summing the load due to each offered traffic flow. Then, the channel assignment algorithm visits each virtual link in decreasing order of expected traffic load and greedily assigns it a channel. In this subsection, we compare the performance of our heuristic channel-width adaptation algorithm with the typical WMN architecture "Hyacinth." In "Hyacinth" architecture, we want to study the impact of different static spectrum partition styles. Specifically, three cases are investigated: (1) The 120 MHz-wide available spectrum is divided into twelve 10 MHz-wide channels. (2) Six 20 MHz-wide channels and (3) Four 30 MHz-wide channels.
The simulation scenario is an area of 1000 m×1000 m consisting of 40 randomly located mesh nodes. Among the 40 nodes, 3 nodes are randomly chosen to act as gateways and 15 nodes are chosen to generate UDP traffic flows towards one of these gateway nodes. The initial rate of traffic flow is also uniformly selected between 1 Mbps and 5 Mbps. Remaining nodes only act as traffic routers. We proportionally change every flow's rate until the network can satisfy 90% of the traffic. In this subsection, both the "Hyacinth" architecture and our algorithms adopt the single-path routing.
6.3. The Impact of Spectrum Block Size
The most important system parameter in our algorithms is the size of spectrum block . With small spectrum block size, we can adjust channel width in a finer granularity and it is possible to obtain more performance improvement. However, using too small spectrum block size will incur significant hardware cost and computation complexity. In this subsection we investigate the impact of spectrum block size on the network performance.
6.4. The Impact of Routing Scheme
6.5. The Impact of Number of Interfaces per Node
In this paper, we address how to adapt channel width to make full use of the spectrum resource in multi-radio multi-channel wireless mesh networks. We mathematically formulate the channel width adaptation, topology control and routing as the mixed 0-1 integer linear optimization. We also propose a heuristic assignment algorithm. Simulation results show that our algorithm can significantly improve spectrum use efficiency and network performance.
Our work distinguishes from prior optimization works in that it does not treat the spectrum as the set of discrete orthogonal channels but the continuous resource. The combination of variable channel widths and center frequencies offers rich possibilities for improving system performance. A lot of things still need to be done. Currently, we are exploiting the partially overlapped channels with adaptable widths in our model to further improve the spectrum efficiency.
Algorithm 1: MILP-based Heuristic Iterative Local Search Algorithms.
3. Run the programming MILP-1 on the full connected networks under the given initial spectrum
(c) Solve the modified programming MILP-1 with the following variables:
while the values of others are kept as constant as in previous iteration. The new objective
(d) Solve the modified programming MILP-2 with the same set of variables as in step 5(c) while
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