 Research Article
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Flow Oriented Channel Assignment for Multiradio Wireless Mesh Networks
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 930414 (2010)
Abstract
We investigate channel assignment for a multichannel wireless mesh network backbone, where each router is equipped with multiple interfaces. Of particular interest is the development of channel assignment heuristics for multiple flows. We present an optimization formulation and then propose two iterative flow oriented heuristics for the conflictfree and interferenceaware cases, respectively. To maximize the aggregate useful endtoend flow rates, both algorithms identify and resolve congestion at instantaneous bottleneck link in each iteration. Then the link rate is optimally allocated among contending flows that share this link by solving a linear programming (LP) problem. A thorough performance evaluation is undertaken as a function of the number of channels and interfaces/node and the number of contending flows. The performance of our algorithm is shown to be significantly superior to best known algorithm in its class in multichannel limited radio scenarios.
1. Introduction
There exists considerable interest in deploying wireless mesh networks (WMNs) as a backhaul in support of ubiquitous broadband wireless access for mobile as well as fixed clients [1, 2]. We study an infrastructure WMN, where a collection of stationary wireless mesh routers are interconnected as a multihop backbone to provide clients, network access. As shown in Figure 1, among all mesh routers, some have client connectivity (mesh access points), and some have Internet gateway capability. The mesh backbone then supports multihop flows amongst mesh routers. These flow traffics are aggregated from a group of clients, leading to needed longterm stability for system optimization.
The success of such WMNs largely hinges on their potential for scaling backbone throughput with increasing client density while preserving network coverage. As has already been noted in [3–5], a crucial element in throughput scaling is more effective utilization of available multiple (nonoverlapping) channels. We propose to achieve this by having multiple interfaces per node at incremental hardware cost; for example, note the availability of integrated 802.11 a/b/g transceivers on the same network interface card (NIC). In the future, such integration of multiple transceivers (such as different generations of a successful technology such as 802.11) on a single device will be commonplace, and our work is thus well aligned with this trend.
Channel assignment (CA) is a resource management challenge in such a multiradio multichannel paradigm, whereby a particular channel is now assigned to a specific link and network interface pair. This is a largely underexplored domain with the potential of boosting network throughput as it can mitigate both intra and interflow interference. Optimal channel assignment would simultaneously involve the choices of routing metric, rate control, power control, and medium access control (MAC). Such an integrated global approach is ambitious and results in very complex systems optimization; typically, only a partial subset of all available resources are used to obtain insight, and our work is no exception.
Various channel assignment algorithms [6–16] which are studied in literature can be classified into packet level, link level and flow level based on their granularity. Packet level algorithms are highly dynamic, where channels are assigned for every packet [6] or every few packets [7]. Channel switching overhead [17] is the major drawback of these algorithms. In flow level algorithms, channels are assigned to several successive links which compose a flow or a set of intersecting flows. In [18], channels are assigned to flows along with the route establishment in a singleradio multichannel network. In [19], links from intersecting flows are assigned the same channels. Flow level algorithms are free of channel switching and are aware of traffic flows however, at the cost of spatial reuse, especially in a dense network. Link level algorithms belong to the most popular category, in which channels are assigned at the granularity of a link. Some optimize linkbased metrics, for example, minimizing accumulated interference among links [8, 9] and maximizing weighted sum of simultaneously active links [10]. In networks with stable topology and traffic flows, endtoend metrics have been shown to be a better suit [13–16]. The performance metric in [13] is the crosssection goodput which is defined as the sum of all useful bandwidth between traffic aggregation nodes and gateway nodes. In [14], the minimum throughput that can be routed from a node to the Internet is maximized. Endtoend throughput optimizations are studied in [15], however, without concrete protocol design. In [16], the authors assume that all links reside in a single collision domain and formulate channel assignment as a game, where players (links) compete for the same pool of channels. Links from a multihop flow form a coalition, and their payoff is the endtoend flow rate.
Both flow oriented heuristics proposed in this paper are both link level algorithms that maximize aggregate endtoend flow rates. The flow oriented graph coloring (FOGC) algorithm is designed for the conflictfree case, and it is extended to a general flow oriented channel assignment (FOCA) heuristic for the interferenceaware case. Both iterative algorithms identify instantaneous bottlenecks of multihop flows based on previous assignment, routing, and topology information. To resolve congestion at each iteration, they first try to assign a new channel to then bottleneck then optimally allocate link rate among a group of contending flows that share this link by solving a linear programming problem. A simulation example in Figure 2 helps demonstrate the impact of bottleneck and the benefit of resolving congestion. In Figure 2, three flows , , and share a common link . Each node has three NICs, and there are four nonoverlapping channels. Let us assume that do not conflict with , and vice versa. Operating under conflictfree condition, links achieve approximately 23 Mbps per channel, while has a rate of approximately 10 Mbps per channel. It can be easily shown that a naive algorithm which simply assigns channels to high data rate links suffers from the congestion at bottleneck even though the overall link rate is maximized. By considering the flow/routing information, a better strategy is to favor the bottleneck even though it has a lower perlink reward. In this simple example, resolving congestion increases the overall network throughput by 100%.
In particular, the key contribution of this work is threefold:

(i)
A generalized optimization model for flow oriented channel assignment in a multiradio multichannel mesh backbone,

(ii)
A flow oriented graph coloring (FOGC) algorithm that maximizes the aggregate useful flow rates for conflictfree channel assignment problem,

(iii)
A centralized flow oriented channel assignment (FOCA) heuristic for interferenceaware cases. It maximizes the aggregate useful flow rates by utilizing crosslayer routing and network topology information.
The remaining of this paper is organized as follows. System model and problem formulation are presented in Section 2. We first introduce and evaluate FOGC algorithm for conflictfree cases in Sections 3 and 4. Then this algorithm is extended to FOCA algorithm in Section 5. Simulation of FOCA is presented in Section 6, followed by concluding remarks in Section 7.
2. System Model and Problem Formulation
We model a wireless mesh backbone consisting of wireless routers (nodes), active links, and multihop flows. The connection between mobile clients and wireless access points is out of the scope of this paper. The sets of wireless routers, links, channels, and flows are indexed as , , and , respectively. Notations , , are used for the th link, the th flow, and the th channel, respectively. For convenience, all notations are summarized in Table 1 with additional explanations as follows.

(i)
Channel Assignment Matrix. for all . An element denotes that link occupies of channel .

(ii)
Routing Matrix. for all , representing the global routing information. If , flow traverses link . A multihop flow is then defined by the set of links it traverses, that is,
(1) 
(iii)
NIC Vector. An element (for all ) is the number of network interface cards on node .

(iv)
Interference Matrix., for all , representing the binary symmetric conflicts among links. Entries in are determined using a binary twohop interference model that is widely used in graph modeling of wireless networks [10, 14, 20] (whereby links within two hops of each other deemed to be mutually interfering).

(v)
Clique Matrix. for all . A clique is a maximum subset of links such that all distinct pairs in this subset would potentially conflict with each other [21]. Assume the total number of cliques is ; each row of , thus represents one of these cliques. Clearly, depends on network topology, and links within one clique must time share a channel if they are assigned the same one:
(2)where symbol denotes componentwise inequality.

(vi)
Channel Reward. for all . Reward is the maximum supportable data rate of link when it exclusively uses channel .

(vii)
Achievable Link Rate Vector.. A rate (for all ) is the achievable data rate of under a given channel assignment decision:
(3)where diag() returns a column vector formed from all the diagonal elements.

(viii)
Flow Rate Vector.. A rate , is the supportable endtoend rate of flow . The total rate of the contending flows traversing a common link cannot exceed the link's rate
(4) 
(ix)
Link Rate Allocation Matrix.. A flow is allowed to use up to of a link 's rate. If does not traverse (), is set to 0. Equation (4) is then expressed in a componentwise way:
(5)Both and are hidden variables in the formulation (7) that we will present soon; however, they will act as important roles in both heuristic algorithms.

(x)
Network Interface Constraint. The total number of channels assigned to links incident to a node cannot exceed the node's NIC number. Channels are assigned to links while NIC constraint resides on nodes. We introduce a network topologydependent matrix to solve such contradiction. The entry denotes that node is one end of link . By mapping the channel assignment results to each wireless router, the NIC constraint is given by:
(6)where the cardinality function Card() counts the number of nonzero entries in each column; hence Card() is the number of NICs needed at wireless routers.

(xi)
Flow Oriented Metric. is a continuously differentiable, increasing, and concave utility function of endtoend flow rates, that is, the aggregated flow rate .
Given the notations above, the flow oriented channel assignment problem in mesh backbone can be stated as the following convexcardinality (nonconvex) problem:
The optimization variables are channel assignment decisions and supportable flow rates . The routing matrices is the required crosslayer routing information, while matrix and are computed from the topology information. Channel reward is considered to be a known parameter. If there always exist sufficient NICs per mesh router, the only cardinality (nonconvex) constraint is eliminated, making (7) a convex problem.
3. Flow Oriented Graph Coloring Algorithm (FOGC)
In some network systems—such as TDMAbased mesh—conflictfree channel assignment is required. Then assigning each channel becomes a binary decision:
and assignment decisions are mutually exclusive among conflicting links:
Substituting (8) for and (9) for in (7), the general formulation is reduced to
When there is no NIC limitation and is linear, (10) is further reduced to an integer linear programming (ILP) problem by removing the last constraint. Optimal solution for ILP problems can be obtained numerically.
Several past work have demonstrated the effectiveness of greedy graph coloring algorithms in solving conflictfree resource assignment problems. A greedy solution, Progressive Minimum Neighbor First (PMNF), is proposed in [22] for the unified time/frequency/code domain channel assignment problem. The PMNF is extended to ColorSensitive Graph Coloring (CSGC) in [20] to solve the channel assignment problem in a multichannel scenario. In the colorsensitive contention graph, vertices are connected via multiple colored edges that represent multiple channels, and colors may have different rewards on different vertices.
Following this colorsensitive graph coloring approach, we present a flow oriented graph coloring (FOGC) algorithm that heuristically solves (10) with the objective function , where is the requested endtoend rate of flow . FOGC is an iterative algorithm that resolves instantaneous congestion. In each iteration, FOGC first identifies bottleneck links and assigns a new channel to the most critical one. This heuristic process is carefully designed so that the conflict constraint (the 4th constraint) and the NIC constraint (the 5th constraint) in (10) are satisfied. Therefore, the remaining process prior to next iteration is an LP problem that optimally allocat the rate of each link to contending flows that share the link. This iterative algorithm is of low complexity, based on local optimality for each additional assignment.
We use a symbol—vector —to store the instantaneous bottleneck links^{1} of all flows. If the endtoend traffic demand of a flow has already been satisfied, there is no bottleneck for that flow. The superscript on all notations denote the current iteration, that is, is 's value in the th iteration.
Step 0 (initiate ).
Assuming no prior knowledge,
Step 1 (identify bottleneck Links).
There is
where is the requested rate of . The set of bottleneck links, , are labeled and sorted in descending order according to their unsatisfied traffic demand. An allzero vector indicates the end of the algorithm, otherwise the link with the largest label is selected. Future steps may set a label value to zero if the rate of that particular link can not be improved by assigning a new channel.
Step 2 (channel assignment strategy (update to)).
Assume that link connecting nodes and is the critical bottleneck selected in Step 1. The channel assignment decision varies according to the number of available NICs on and .

(1)
If and both have free NICs and at least one available channel in common, assign the most beneficial^{2} available channel in common to link . Eliminate from the available channel list of all 's and 's conflicting neighbors.

(2)
Otherwise, redirect to Step 1, set the label value of to zero, and reselect a link with the largest label.
Step 3 (link rate allocation (solve )).
Entering with the th channel assignment matrix , we next calculate the link rate allocation matrix and the supportable flow rate vector via optimally solving the following LP problem:
Thus far, we have a new link rate allocation matrix . Returning to Step 1, we can determine a new set of bottleneck links for next iteration. This algorithm terminates when an allzero label vector appears in Step 1.
Since the 4th and 5th constraints in (10) are satisfied in the heuristical process in FOGC, the optimal result from (13) is not necessarily an optimal solution for (10). The following proposition shows that FOGC achieves optimality in a simple single flow case.
Proposition 1.
In the single flow case, FOGC is the optimal channel assignment when channels have identical reward.
Proof.
Assume that is an optimal channel assignment that assigns a channel to link at its th operation. It is easy to verify that exchange the sequence of terms in does not affect its optimality. Let be the assignment from FOGC algorithm which has exactly the same first assignments as . Here we will show that either agrees with the optimal assignment on its first assignments or has already achieved optimality on the th assignment. If and , swap and in . Then has the same first assignments as . If and , then the flow rate will never increase after the th assignment in , because is an instantaneous bottleneck which is never treated again in . Then achieves optimality at its th operation.
In the simplest case, FOGC's locally optimal strategy leads to global optimality. In general, FOGC's performance in more complicated scenarios is evaluated through simulation in the following section.
4. Simulation Results of FOGC
Though FOGC algorithm works for any network system where conflictfree assignment is desired and the interference among links can be modeled using distancebased conflict graph, for demonstration purpose, FOGC algorithm is simulated in ring and grid topologies (Figure 3) using ideal twohop interference model. FOGC is compared with two other algorithms. The baseline is a centralized LinkBased Graph Coloring (LBGC) algorithm which seeks to maximize the overall link rates without routing knowledge. The upper bound is the optimal channel assignment^{3} (OPT) obtained by exhaustive search. In LBGC, the flows traversing a common link share this link's rate fairly.
In an node ring, each link only conflicts with adjacent 4 links. Two hop flows are randomly generated. Assume that each node has sufficient NICs and the number of available channels is varied from 2 to 8. An immediate observation from Figure 4 is that no endtoend throughput is obtained when there are less than 3 channels. This is simply due to the conflictfree constraint and the predefined 2hop interference model. As one can see, FOGC follows OPT closely and outperforms LBGC 120% in the case of 8 channels. This demonstrates that resolving instantaneous bottlenecks is a good greedy heuristic to achieve near optimal performance at low complexity.
In an grid topology, several sourcedestination pairs are randomly selected, and the shortest path is randomly selected for each pair. Up to 17, identical channels are provided to three 8hop flows. As Figure 5 shows, with sufficient NICs, the average endtoend flow rate scales almost linearly with channel resources in presence of 3 or more channels. When the number of NICs on each node is limited, the per flow throughput would saturate at some stage. This saturation point reveals the resource utilization efficiency of each algorithm. With only 2 NICs, the FOGC almost fully exploits 4 channels and could partially utilize up to 8 channels. Overall, Figure 5 demonstrates that FOGC is superior to LBGC in terms of efficiency in utilizing multiple channels with very limited number of NICs, by using additional routing and network topology information.
5. Flow Oriented Channel Assignment Algorithm (FOCA)
Due to the broadcast nature of wireless communication, the conflictfree assumption is rarely true in practice and should be viewed as a model approximation. The a forementioned graph coloring algorithm FOGC is extended to a general flow oriented channel assignment (FOCA) algorithm to heuristically solve (7). Conflicting links are allowed to timeshare the same channel, and in each iteration, apart from assigning a new channel, channel switching mechanism and channel switching propagation are designed to adjust existing assignment decisions.
FOCA has a similar iteration process as FOGC. It first identifies the bottleneck links, then assigns a new channel, or adjusts existing channel assignment to resolve congestion, and prior to entering the next iteration, optimal allocation of link rate to contending flows that share this link is computed by solving an LP problem.
Step 0.
Same as FOGC.
Step 1.
Same as FOGC.
Step 2 (channel assignment strategy (update to)).
Assume that link connecting nodes and is the critical bottleneck selected by Step 1. The channel assignment decision varies according to the number of available NICs on and .

(1)
At least one free NIC on both and : assign a locally least occupied channel to a free NIC on and , respectively.

(2)
No free NIC on at least one free NIC on (or vice versa): choose locally least occupied channel from 's channel usage list and have one free NIC of tuned to that channel to communicate with .

(3)
Neither nor has any free NIC:

(a)
and have at least one common channel^{4}: communicate on a least occupied common channel utilizing the NICs already tuned to that channel.

(b)
and have no common channel, and the connectivity between and has already been satisfied: return to Step 1 to set the label of link to zero.

(c)
and have no common channel in use, and the connectivity between and is not satisfied: channel switching and channel switching propagation are needed. A least occupied channel is selected from 's and 's channel usage list, respectively. Assume they are and , and carries a lower traffic load. As Figure 6 shows, 's NIC that originally works on is switched to to communicate with . To preserve 's connection with its own neighbors on , this switching is propagated to those neighbors, and then to neighbors of those neighbors, and so on.

(a)
At the completion of this step, we have updated the channel assignment matrix to .
Step 3.
Same LP optimization as in FOGC.
6. Simulation Results of FOCA
The flow oriented channel assignment (FOCA) is simulated using random network topologies (Figure 7) in ns3 [23] simulator because (i) node model is architected to support multiple interfaces, (ii) channel model is architected to support multiple channels in ns3. Customized functions are implemented to install multiinterface on nodes and bind interfaces to channels according to different channel assignment algorithms^{5}. First, we generate random topologies, multiple sourcedestination pairs, and one least hopcount path for each pair. This routing information is stored to assist future channel assignments. Then channel assignment algorithm is performed in a centralized manner before data transmission. All results are averaged over multiple runs on multiple random topologies. The FOCA is compared to a unified onetoone channel assignment, which assigns the th channel to the th NIC. The unified algorithm is of zero complexity and performs well when the number of NICs matches with the number of channels. It generates a much better baseline than any random assignment, which is unfortunately always used as an inaccurate benchmark. Another contrast is an empirical traffic loadaware channel assignment (LACA). In [13], LACA, a routing algorithm, as well as the control loop between them are designed. Assumeing a precalculated routing and fixed injected traffic load, the LACA assigns channel resources to links in the decreasing order of the expected link load. The original evaluation metric "crosssection bandwidth" is similar to the endtoend flow rate in this paper.
In realworld applications, the NIC limitation is frequently confronted before the number of channels becomes a limitation. Therefore the multichannel limitedradio scenario performance is the key criteria in evaluating channel assignment algorithms. We first simulate three 5hop flows in random networks, where each node has only 2 NICs, and vary the number of channels. Similar to the conclusion from Figure 5, in Figure 8, FOCA's throughput saturates at approximately 9 channels and outperforms LACA approximated 50% in the saturated case. Once saturation is reached, increasing the number of channels causes only jitter but not significant increment in flow rate, because now the limitation is insufficient NICs on some bottleneck links. A good channel assignment algorithm will efficiently utilize as many channels as possible with very few NICs by diversifying traffic over as many channels as possible to avoid intra and inter flow interference. Hence, the saturation point (in terms of the number of channels) and the saturation flow rate in the multichannel limitedradio scenario are of particular interest to system designers.
Next we evaluate the performance when sufficient NICs are provided. Three 5hop flows are injected in random networks with 6 available channels while the number of NICs per node is varied. As one could see from Figure 9, with sufficient NICs per node, the simplest unified onetoone channel assignment (baseline) will outperform both FOCA and LACA. The region in which FOCA (LACA) stays above the baseline is FOCA's (LACA's) effective operating region. For example, with these particular parameter settings, LACA should be avoided when there are 4 or more NICs per node, and FOCA is a favorable algorithm only up to 5 NICs per node. Figure 9 clearly justifies that both FOCA and LACA are designed for multichannel limitedradio scenarios.
Scalability is a practical challenge for mesh backbone. We simulate the aggregated flow rate as well as the per flow rate by gradually increasing the number of coexisting 3hop flows. The random networks simulated have 6 channels and 2 NICs per node. As expected, the per flow throughput in Figure 10 decreases steadily as the number of coexisting flows increases. However, as shown in Figure 11 shows, the aggregated flow throughput increases with diminishing slope till the network becomes saturated. FOCA has a higher saturated aggregate throughput due to its effective resource management. Further increasing the amount of coexisting flows leads to more variance in FOCA than in LACA. Fortunately, no significant aggregate throughput drop is observed, showing that FOCA has the acceptable scalability when network becomes overloaded.
7. Conclusion
The fast deployment of WMNs places stringent requirements on endtoend rates of the underlying multihop mesh backbone, especially in large scale and dense scenarios. This paper studies the channel assignment problem for a multichannel multiradio mesh backbone. We first presented a general optimization formulation, which is nonconvex due to practical constraints. Then a flow oriented graph coloring (FOGC) greedy algorithm was designed for the conflictfree case, and it was extended to a flow oriented channel assignment (FOCA) heuristic for the general case where interference is considered. Both algorithms identify instantaneous bottlenecks at each additional assignment iteration using routing and network topology information and try to resolve congestion by (i) assigning a new channel, (ii) channel switching (FOCA only), and (iii) optimally allocating link rate to contending flows by solving an LP problem. Simulation not only demonstrated the effectiveness of addressing bottlenecks at each additional iteration but also showed that both FOCA and FOGC can efficiently exploit multiple channels with very limited number of NICs.
Although both heuristics are for a static network, their low complexity as well as the relatively high stability of traffic flows in mesh backbone allow the channel assignment decision to be adapted periodically or once the accumulated changes exceed a certain threshold. While the proposed algorithms pursue local optimality at each additional assignment, this does not lead to global optimality. An option is to extend this heuristic (at cost of additional complexity) by looking several steps ahead, that is, considering more than one critical bottleneck link at each iteration. This framework can be applied towards other utility functions as well.
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Endnotes
1.In case that a multihop flow has several bottleneck links, one link is randomly selected to break the tie.2. The most beneficial channel is the one which provides the highest reward (data rate).3. Since the complexity of exhaustive search scales exponentially with the number of links and channels, we only present OPT in a ring topology.4. A common channel is the channel that both and are using to communicate with neighbors other than each other.5.Simulation code available at http://www.ee.washington.edu/research/funlab/mrmc/index.html.
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Keywords
 Channel Assignment
 Wireless Mesh Network
 Mesh Router
 Link Rate
 Bottleneck Link