Rematch: a highly reliable scheduling algorithm on ... - Springer Link

J Supercomput (2010) 51: 224–240 DOI 10.1007/s11227-009-0285-6

Rematch: a highly reliable scheduling algorithm on heterogeneous wireless mesh network Panlong Yang · Guihai Chen

Published online: 24 April 2009 © Springer Science+Business Media, LLC 2009

Abstract In highly dynamic and heterogeneous wireless mesh networks (WMN), link quality will seriously affect network performance. Two challenges hinder us from achieving a highly efficient WMN. One is the channel dynamics. As in real network deployment, channel qualities are changing over time, which would seriously affect network bandwidth and reliability. Existing works are limited to the assumption that link quality values are fixed, and optimal scheduling algorithms are working on the fixed values, which would inevitably suffer from the link quality dynamics. Another challenge is the channel diversity. In single channel wireless networks, channel assignment and scheduling are N P-hard. And in multichannel wireless networks, it could be even harder for higher throughput and efficient scheduling. In this study, we firstly characterize the stochastic behavior on wireless communications in a Markov process, which is based on statistical methodology. Secondly, on exploiting the stochastic behavior on wireless channels, we propose a stochastic programming model in achieving maximized network utilization. Considering the N P-hardness, we propose a heuristic solution for it. The key idea in the proposed algorithm is a two-stage matching process named “Rematch.” Indeed, our solution to the stochastic network scheduling is a cross-layer approach. Also, we have proved that it is 2-approximate to the optimal result. Moreover, extensive simulations have been done, showing the efficiency of “Rematch” in highly dynamic and distributed wireless mesh networks. Keywords Wireless mesh network · Heuristic · Matching algorithm · Distributed algorithm · Stochastic programming P. Yang () · G. Chen Institute of Communication Engineering, PLAUST, Department of Computer Science, Nanjing University, Hankou Road, Nanjing, Jiangsu Province, People’s Republic of China e-mail: [email protected] G. Chen e-mail: [email protected]

Rematch: a highly reliable scheduling algorithm on heterogeneous

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1 Introduction The emergence of wireless mesh networks (WMN) are pervasive in providing Internet access service for mobile devices. Mesh routers, for example, are taking advantage of multiple radio interfaces, such that the improved network throughput can be achieved. In order to build a more pervasive and agile WMN, two challenges need to be properly dealt with. The first challenge is the channel state variations. In wireless communications, channel quality would possibly be affected by geomagnetic conditions, signal propagation effects, and weather conditions. During the packet transmission time, link quality variations would undergo a series of possible values, which are sustaining for different periods. Although link dynamics and the distribution can be available via statistical method, it is always assumed that for a long period, channel quality value distribution is relatively stable [11]. Unstable channel quality would make the previously proposed link schedule and channel assignment undergo very low efficiency. The second challenge lies in heterogeneity, where in WMN, transmission power, bandwidth, and available channels are all different to each other. With the increasing number of channels and radio interfaces, the problem complexity also increases dramatically. Many research works have been done on the MCMR WMN scheduling algorithm. For example, [1, 7] explore the upper bound of wireless networks, and the lower bound is achievable as the effective scheduling algorithm is applied. Also, some works jointly consider the channel assignment and routing algorithm in achieving an optimized throughput [3]. Although capacity bounding techniques, indeed, generally work on an ideally fixed UDG (Unit Dist Graph) model, it seldom exists in real network deployment. In [2, 4], channel assignment and link scheduling algorithms are jointly studied in a system view, where routing and MAC are considered as interactive layers are in a cross-layer design. Considering cross-layering effects, some of the seminar works on optimizations jointly consider channel assignment, routing, power control, and route selection [6] and improve network performance dramatically. Some of the studies are concerned on bandwidth allocation in an optimized manner given traffic demand [6, 9]. All these works can be categorized as static scheduling, because they all assume a fixed network topology, reliable wireless communication channels, and even traffic patterns. In this study, we make an investigation on the scheduling algorithm of MCMR WMN in dynamic channels. Our algorithm is based on a cross-layer approach, where stochastic physical layer information are used for throughput maximization. Contributions of this paper are listed as follows: 1. Rematch Method and the Stochastic Programming Model: We proposed a stochastic programming model. In our stochastic programming model, channel parameters are stochastic and the distributions of channel values are presented by a consequence of values. A two-stage matching algorithm is proposed in dealing with the stochastic programming difficulty. And our method is called “Rematch,” as it is a two-stage matching process in achieving a maximized throughput. 2. Method on Achieving Channel Quality Values: We propose a statistical method in achieving the channel quality values during a period of time. In order to save the probing cost and keep the estimation accuracy at the same time, we propose a

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smooth merging algorithm. An expectation maximization (EM) method is applied in dealing with the loss of data. Our proposed method can adaptively measure the channel quality distribution in an efficient way. 3. Heuristic and Distributed Scheduling Algorithm: We propose a heuristic algorithm in dealing with the N P hardness of the scheduling problem among multiple channels. The heuristic however, as we have proved, is “2-approximate” to the optimal value. Our proposed scheduling algorithm can be executed in a distributed way, which is important to the WMN environment. The remainder of this paper is organized as follows: In Sect. 2, related work is introduced for background knowledge. In Sect. 3, we present the system model and problem formation in description of the problem. In Sect. 4, we make a thorough analysis and model our problem and algorithm on computing the channel quality distribution is introduced in Sect. 5. Algorithm description and performance analysis on “Rematch” are introduced in Sect. 6 and Sect. 7, respectively. Finally, in Sect. 8, we draw our conclusions and present the future research work.

2 Related work Recently, most of the seminar works are focusing on the capacity upper bound for MCMR WMN, which is estimated according to the static and dynamic link channel assignment [2]. Moreover, Alicherry et al. [3] have proposed a conjunctive work on channel assignment and routing optimization algorithm, unlike ours, that study focuses on the infrastructure-based wireless mesh networks where a central base station could effectively schedule transmissions among communication regions. In [7, 8], Kyasanur and Vaidya present the capacity of multichannel wireless networks scale with respect to the number of orthogonal channels. Since the radio interface and channels are not available as theoretical results have assumed, it would not be applicable to MCMR WMN with a limited number of channels. Min et al. make efforts in providing QoS in wireless networks [27], and also [26] proposes a framework for heterogeneous wireless network. A distributed algorithm is proposed in [6] for spectrum allocation, power control, routing, and congestion control in wireless networks. Given a feasible spectrum allocation, the algorithm can effectively control the transmission powers. However, they assume that the spectrum can be divided into sufficient number of subchannels as well. As to our models, we need not this assumption, which is more generalized and realistic than ever. An interference-aware channel assignment is also in mesh routers and colocated wireless networks [4]. The proposed algorithm BFS-CA is a dynamic, interference-aware channel assignment algorithm. Scheduling of wireless data is considered in multicarrier wireless data systems [5]. Different carriers have channel rates separately, and a scheduling algorithm is needed in order to enhance the performance of WiMax access point. Unlike ours, they only consider the one-hop scenario, and it is a data scheduling algorithm instead of a time-slot scheduling algorithm as [1] has proposed. Efforts are also needed before finding an opportunistically optimal channel and time period to transmit packets. As mentioned in [12], it provides transmitters more


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channels, and software defined cognitive radio systems strengthen the multichannel systems, which proves to be more intelligent by applying programmable paradigm embedded in hardware. Software radio system can intelligently tune the frequency bands and select among different modulation techniques, aiming at selecting the optimal frequency for transmissions [13]. The cognitive radio system, however, provides a newly built way to use radio frequencies. Due to the collaboration in network environment, the system can make a better usage on network resources for its users [14]. Many seminar works have been proposed in order to make a better utilization on the time-varying channels [15–17]. Seminar work in [16] presents a pure threshold based algorithm on optimal stopping strategy of channel probing. It is a tradeoff between channel quality reward and probing cost. Zheng et al. [17] propose a distributed scheme, which models the probing procedure as a contention and threshold based process. Seminar work in [16] assumes that channel state is independent of the state of other channels during transmission. This is true if we take the channel state as the only metric for opportunistic transmission. However, in real network deployment, if the channels are not enough, the links in the network would congest for a limited number of channels. Selecting one channel would face challenges. One is that the larger reward channels would possibly be preferred by all channels and the correlation between channels and node deployment [16] would make the contention even worse. Another challenge is that, as the time progresses, more channels are probed, and the rest of the channels unused by other transmissions are becoming useless, which would worsen the contention on channel selection. In this work, we keep the channel dynamics in mind during the algorithm design process, where stochastic behavior would lead to a totally different design on channel assignment and scheduling. Ours is different from conventional methods, which are purely trying to achieve the optimal value. In wireless networks, we cannot do as much complicated work on computing and coordination as we do in the Internet community [23–25]. Indeed, our proposed algorithm need only statistical results on channel variations, which could be easily achieved during data transmissions, and need not any additional overheads. Moreover, ours can also work with the cognitive scheme. In the following section, we will build the system model firstly for problem modeling and algorithm design.

3 System model In this section, we introduce the network model and wireless channel model. Multichannel multiradio communications are properly defined and the MAC layer interference is also mathematically modeled for further analysis. We propose a Markovian channel model, where state variations are modeled according to the temporal behavior. 3.1 Network model We assume that there is a set V of mesh routers deployed in a plane. Each node vi ∈ V denotes a mesh router in network G = {V , E T }, and we will illustrate the

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Fig. 1 Communication resource demonstration of MCMR routers

extended edge set E T in the following paragraph. Let Q(vi ) denote the number of radio interfaces on mesh router vi . Considering MIMO and OFDM technology, each router can effectively merge and split wireless bandwidth of each radio interface, and the number of orthogonal channels on interface ϑk of node vi is O(vi , ϑk ). The notation 1≤k≤Q(vi ) O(vi , ϑk ) denotes the total number of orthogonal channels of node vi , while the orthogonal channel set of node vi on interface ϑk can be denoted as C(vi , ϑk , λj ), where 1 ≤ k ≤ Q(vi ), and 1 ≤ j ≤ O(vi , ϑk ). Each node vi uses a transmission range t (vi , ϑk ) on interface ϑk to send data. As shown in Fig. 1, the bandwidth on interface ϑk can be denoted as Bk = {b1k , b2k , . . . , bO(vi ,ϑk ),k }, where the elements in the set bi k are the values of bandwidth on channel of interface ϑk . iO(v ,ϑ ) Bk denotes the total bandwidth on interface ϑk , where Bk = i=1 i k bi k . On node vi , the transmission range Rk ∝ t (vi , ϑk ), and {Rk , Bk } characterize the heterogeneous network on radio interfaces. e(vi , vj , ϑk ) ∈ E T denotes the extended undirected edge between node vi and vj on interface ϑk . If the following condition is satisfied, there exists an edge e(vi , vj , ϑk ) ∈ E T between vi and vj : vi − vj ≤ min t (vi , ϑk ), t (vj , ϑk ) The notation vi − vj denotes the Euclidean distance between node vi and vj . Note that the number of radio interfaces would be identical to each other, which can be denoted as ϑk . Concurrent transmissions between neighbor nodes are available if and only if the following two conditions are satisfied: – Two transmissions tri and tri are not using the same radio interfaces, which is dej noted as (tri ) = I (tr ). – Two transmissions belong to the same interface but they belong to different orthogj j onal wireless channels, that is, I (tri ) = I (tr ), and OC(tri ) = OC(tr ). Interference models on WMN generally fall into three types, which are proposed as the Protocol Interference Model (PrIM), Fixed Power Protocol Interferences


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Model, and RTS/CTS Model [1]. In this study, we mainly focus on the RTS/CTS model, since it is widely applied in WMN. Our scheduling algorithm is relatively independent to interference models, and can be applicable to other interference models mentioned above with moderately modifications. 3.2 Link quality model Wireless communications channel states are varying with time, as shown in Fig. 2, the transmission rate is constantly changing over the time. In this study, we mainly map the channel states to the channel bandwidth. The SNR based analysis is out of the scope of this paper. A thoroughly study on SNR and bandwidth is proposed in [19], which would help us build a more reasonable channel states model. As shown in Fig. 3, on each state i, the transmission rate, that is, the bandwidth equals ei . We build our channel quality model based on the following rules: 1. Let ei,j (t) denote the available bandwidth of link ei,j at time t. If the sampling interval is sufficiently small, given any sampling interval τ , the channel bandwidth is constant. 2. And we also assume that, under the sampling rule, the wireless channels will endure series of channel bandwidth E c = {e1 , e2 , . . . , em }, and the cardinality of E c is finite. 3. Channel dynamics can be modeled into the state transition process. For each value ei , the probability of transition to ej at next sampling interval is Pi,j , and also, m j =0 Pi,j = 1.

Fig. 2 Illustration on channel state dynamics with time progresses

Fig. 3 Illustration on the channel state transition model

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The transition matrix is given Fig. 3. ⎛ P0,0 ⎜ P1,0 ⎜ TE =⎜ ⎜ P2,0 ⎝ ... Pm,0

by T E , and the state transition model is shown in P0,1 P1,1 P2,1 ... ...

P0,2 ... ... ... ...

... ... ... ... Pm−1,m

⎞ P0,m P1,m ⎟ ⎟ P2,m ⎟ ⎟. ... ⎠ Pm,m

The channel quality and state transition model makes the stochastic programming model on maximized throughput in dynamic network possible. We will present the formation of our problem and propose the stochastic programming model in Sect. 4. The dynamic behavior, the probability of state transition, and the bandwidth value of each state, are not available yet. We will solve this problem in Sect. 5.

4 Problem formation On node vi , the transmission power on interface ϑk is denoted as t (vi , ϑk ). We rank the interfaces by transmission power on every node in network with descending order. It can be listed as ϑ (1) , ϑ (2) , . . . , ϑ (Q) . Generally speaking, the number of power level is more than that of interfaces equipped on nodes in WMN. The interface with maximum power level is denoted as ϑ (1) , and the lowest power level is denoted as ϑ (Q) , the ranking list can be denoted as t (·, ϑ (1) ) ≥ t (·, ϑ (2) ) ≥ · · · ≥ t (·, ϑ (Q) ). We use the conflict graph FG proposed in [1] to represent the interference in WMN graph. In our network model, each vertex of FG corresponds to an edge e(vi , vj , ϑk ) ∈ E T in WMN, which is denoted by l(vi , vj ). Subgraph E T (ϑ (i) ) is the network graph formed by edges of the same interface, such that e(vi , vj , ϑ (i) ) ∈ E T (ϑ (i) ), where ϑ (i) is the ith power level interface. Given any two different level of interface ϑ (i) and ϑ (i) , if i < j , and e(vi , vj , (j ϑ ) ) ∈ E T (ϑ (j ) ), a link exists between node i and j on interface ϑ (i) , that is, e(vi , vj , ϑ (i) ) ∈ E T (ϑ (i) ). Channel assignment ς assigns each node vi with a channel in the available orthogonal channels set. Let ς(vi ) denotes the set of different channels assigned to the node, which are selected among all the available channels of node, ranging from 1 to Q(vi ). The first goal of our scheduling algorithm is to propose a valid interference-free channel assignment and schedule among nodes in WMN. The second goal is to maximize the available bandwidth for each node. Since it is a WMN graph, there are two levels of research issues to consider in dealing with the two goals mentioned above. The first level is channel assignment, and the second level is link scheduling. In single channel wireless networks, there is no need to do channel assignment. If the links on interference graph F G are using the same channel, we have to schedule the link on the same channel in order to avoid the collision. Given a network graph G = V , E , and weight of each edge, denoted by ce , the optimal assignment can be achieved if:

max ce xe

xe ≤ 1, ∀v ∈ V ; xe ∈ {0, 1}, ∀e ∈ E . e∈E

e∈δ(v)


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Considering the stochastic programming cases, a first stage edge weight ce , which is the probed channel quality value. And a second stage weight, which is a discretely distributed value, and each edge weight is ranging values in channel quality value set de1 , de2 , . . . , der , with corresponding probabilities denoted by p1 , p2 , . . . , pr . max

ce xe +

r

ps

1

e∈E

des yes

e∈E

subject to xe + yes ≤ 1 ∀v ∈ V , s = 1, 2, . . . , r, e∈δ(v)

e∈δ(v)

xe ∈ {0, 1},

yes ∈ {0, 1},

∀e ∈ E, s = 1, 2, . . . , r.

Theorem 1 The two-stage stochastic matching is N P-complete. Proof The theorem has been proved in [18].

5 Achieving channel quality distributions In this section, we propose an adaptive method. Firstly, we need to deal with limited data sets before characterizing the channel quality distribution. Secondly, we are to calculate channel distribution with series of values probed previously. 5.1 Dealing with limited data For the packet loss rate ei , we are however challenged with lack of sufficient data for confidentially statistical results. We use the expectation maximization (EM) method to overcome the deficiency. In dealing with incomplete data, let y denote incomplete data consisting of values on observable variables, that is, our observed packet loss rate and let z denote the missing observations. An EM algorithm iteratively improves an initial estimate θ0 by constructing new estimates θ1 , θ2 , and so on. It can be computed according to the following equation: θn+1 = arg max D(θ ), θ

where D(θ ) is the expected value of the log-likely hood. D is given by D(θ ) =

p(z|y, θn ) log p(y, z|θ ).

z

Through the iteration process, the value of pi can converge to a steady state with maximal likelihood.

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5.2 Calculating the channel value distribution i

At probing time point t, channel quality value is Qi (t). P [ai ≤ Qi (t) ≤ ai+1 ] = N N, where N is the total number of probing times and N i is the total number of times that Qi (t) falls into interval [ai , ai+1 ]. As we do not know channel quality distribution previously, the interval length and total number are difficult to set. If the interval length is too small, P [ai ≤ Qi (t) ≤ ai+1 ] would be accordingly small, and the total number of intervals are too large, which would lead to computational difficulty and the difference between channel quality is smoothed by so many interval steps. As the interval length is too large, it would lead to inaccurate statistical channel quality distribution. Without loss of generality that we can assume, there are totally r˜ possible intervals for us to handle. The handling process is called “Merge and Split.” We will merge the consecutive interval INTV{i} = {ai−1 , ai } and INTV{i + 1} = {ai , ai+1 } if the values in the interval INTV{i} and INTV{i + 1} are close enough. The splitting process will be applied if the gaps between values of the interval INT{i} are large enough. In order to solve this problem, we propose an interval adaptation algorithm. The algorithm works as follows: If P [ai ≤ Qi (t) ≤ ai+1 ] > α 1r˜ , we should split the interval [ai , ai+1 ] into two 1 , we intervals with [ai , ai +a2 i+1 ] and [ ai +a2 i+1 , ai+1 ]. If P [ai ≤ Qi ≤ ai +a2 i+1 ] > α r˜ +1 ai +ai+1 1 do the same interval-split process until P [ai ≤ Qi ≤ 2 ] < α r˜ +1 . If P [ai−1 ≤ Qi (t) ≤ ai ] < β 1r˜ or P [ai ≤ Qi (t) ≤ ai+1 ] < β 1r˜ , we should merge the interval [ai−1 , ai ] and [ai , ai+1 ] into one interval with [ai−1 , ai+1 ]. We do the 1 same interval-merge process until P [ai ≤ Qi ≤ ai+1 ] > β r˜ +1 . The selections of value α and β depend on the distribution of channel value and the maximum allowed number of value samples. In this paper, we select the value α = 2 and β = 0.5.

6 Algorithm description on “Rematch” In this section, we firstly present the overview of the proposed heuristic algorithm “Rematch.” After that, we introduce the distributed maximum matching algorithm. Finally, we make an analysis on our algorithm for the approximation rate and the complexity. 6.1 Overview of the “Rematching” Our heuristic algorithm is based on the maximum bipartite matching algorithm. It constructs a conflict graph F G , which is properly defined in [1], and each link fij = {li , lj } ∈ F G ; the links li and lj are conflicting to each other. We make a transformation, where the link set E T and channel set I are two node sets on a bipartite graph. If a radio channel is available to link, there is an edge on the bipartite graph. Neighboring nodes on the conflict graph cannot use the same radio interface. In our bipartite matching algorithm, only one channel of an interface in a matching set is


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Algorithm Two-stage Matching Algorithm 1: Compute First-stage Maximum Matching results, and the allocation factor x 2: for i := 1 to s do do 3: Compute Second-stage Maximum Matching results, and zs = d s y s r s 4: z˜ = max z, s=1 ps z 5: end for 6: if z˜ = z then 7: The first-stage result 8: else 9: The second-stage result 10: end if Fig. 4 Two-stage matching algorithm description

Algorithm Graph Conversion 1: Each node v compute the number of neighbors N (v) and send message N (v) out 2: Listen and receive messages 3: if N (v) > N (u) then 4: Direct the edge v to u 5: end if 6: if N (v) = N (u) then 7: Random Select v or u and Direct the select one to the other 8: N (v) +

9: end if 10: if N (v) < N (u) then 11: Direct edge u to v 12: end if Fig. 5 Graph conversion pseudo-code

assigned to the link. The algorithm terminates if all the links in the network are assigned a channel to transmit data and all the channels are assigned to links in the network. s Let x be a first-stage solution, and let z = cx. For scenario r s, let sy be a seconds s s stage myopic solution, and let z = d y . Let z˜ = max z, s=1 p s z . If z˜ = z, then return and (x, 0, 0, . . . , 0); otherwise, return (0, y 1 , . . . , y r ) and rs=1 ps zs . Pseudo codes of a two-stage matching algorithm are listed in Fig. 4. 6.2 The distributed matching algorithm Conversion process and tree decomposition algorithms are referred to similar works in [20, 21], which assume a different ID should be assigned to each router. The description on graph conversion process is shown in Fig. 5. But in real networks, especially WMN, it is very hard to make this kind of network management operation before network deployment in a highly dynamic network. Ours need not assign each

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Algorithm Network Partition Approach 1: Each node sends its calculated number of neighbors N (v) to all neighbors. 2: Order the edges by decreasing N (v) value, which are the candidate proposed rank. 3: Listen and receive messages. 4: The edge value of rank is determined by the end node with higher N (v). 5: Output the tree set with determined rank value i, which form the part of tree, Fi . Fig. 6 Network partition algorithm pseudo code

Algorithm Maximum Weighted Matching Algorithm 1: Compute the Δ + 1-vertex coloring of each Fi . 2: Let ci be the color of Fi 3: M = ∅ 4: for i := 1 to Δ do do 5: for c := 1 to Δ + 1 do do 6: every u such that ci (u) = c selects the most weight one of its outgoing edges; let Mc be the set of edges so selected 7: M = M ∪ Mc 8: Remove all vertices of Mc from the graph 9: end for 10: end for Fig. 7 Weighted maximum matching algorithm pseudo-code

node with different ID. In order to void loop, modify the N (v) value with a moderately small factor , where 1. Our tree decomposition algorithm is based on the number of neighbors and the ranking list among them, such that each induced link is assigned a candidate “color” to partition the tree into a set of trees from F1 to FΔ , where Δ is the maximum degree of nodes in network. The graph partition algorithm and maximum weighted matching algorithm are shown in Figs. 6 and 7 respectively. 6.3 Complexity analysis on “Rematch” We decompose the network overhead of the p-stable algorithm into several processes and compute the network overhead separately. In the graph conversion process, each node would send “HELLO” or “BEACON” message in order to inform the connectivity to its neighbor nodes. O(n) messages need to be sent in the graph conversion process. In the maximum matching process, in a single MWM phase, matching overhead is O(Δ × n), because the embedded “for” cycle is of complexity O(Δ2 ). The nodes on n n ). So, the overhead is O(Δ2 ) × ( Δ ) = O(Δ × n). each decomposed tree Fi is O( Δ 2 The coloring message overhead is O(Δ + log n) [1, 20, 22] and the total messages overhead are O(Δ × (Δ + n) + log n).


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Fig. 8 Illustration on two cases of lost weight on graph

Since we make the graph conversion process on original multigraph G = V , E T to the directed tree T , some edges on the graph causing loops in the graph are deleted, which could eventually lead to the weight lost. We have to consider these lost edges, and schedule transmission on these edges without violating the maximum weighted matching results. There are also two cases as the lost interfaces are considered in adjusting procedure. In the first case, as shown in the left part of Fig. 8, the edge between vj and vk is deleted in building decomposed trees. Considering the reorganizing process, since node vi , node vj , and node vk are at the one hop transmitting range, there are also two cases for scheduling the deleted edge vj , vk . The first case is that channels on edge vj , vk are orthogonal to channels on vi , vj and vi , vk . Thus, the transmission on edge vj , vk will not violate the matching results. The second case is that vj , vk has the same channels with edge vi , vj and vi , vk . Since three nodes are in the one hop transmission range, the transmission schedule can be easily shared. The second case, as shown in the right part of Fig. 8, there are two paths crossing at a node, and there are disjoint paths between the root and the first time crossing node. The edge between vi and vj is deleted. Node vi and node vj can also share the schedule with its neighbors, respectively, as we have shown in the first case. In lost weight process, there would be O(n2 − n) edges that need to be rescheduled in the worst case, while in the average case, it should be O(nα − n), where 1 < α < 2. The total overhead of thep-stable algorithm is the sum of messages overhead in different processes, that is, O(Δ × (Δ + n) + nα + log n). Next, we will analyze the time complexity of the p-stable algorithm. In the process of graph conversion, each node should wait and listen to all of the neighbor’s information, thus the time complexity is O(Δ2 ). According to results in [20], there could be O(Δ + log∗ n) rounds for maximum matching. In each round, the coloring time complexity is O(Δ2 + log∗ n). Thus, the time complexity is O(Δ3 + (log∗ n)2 ). Coloring stage and weighted maximum matching stage are using the heuristic method in dealing with the NP-hard problem. Thus, there could be a gap between the heuristic result and the optimal result. Approximate ratio analysis is to evaluate this gap, and make a further study on our proposed heuristic algorithm. According to the result in [1], the distributed coloring algorithm for the RTS/CTS model computes a valid coloring using at most φ(FGD2 ), where φ(FGD2 ) is the conflict factor of the most conflict edge in G. In the worst case, it is Δ to the optimal result. As to the unweighted maximum matching process, in the worst case, it would lead to a factor 2 to the optimal result. That is, H w ≥ 12 wmax . But in the weighted case, it heavily relies on the distribution of the weight assigned to different channels.

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Given the two phase of weighted matching in S, which can be denoted as S i and w.l.g., i < j . According to our heuristic algorithm, the most weighted element in phase i is larger than that of phase j , and it can be denoted as max{S i } > max{S j }. Assuming the reliability value is uniformly distributed between [Rmin , Rmax ], the min . According to [21], the sum average reliability in the network would be Rmax −R n of weights on all edges with probability is lower than the average level, that is, Rmax −Rmin , which is less than half of the weight on heaviest edges. On average, n there would be one-half with higher reliability above average level, and one-half with smaller ones. Then on each phase, the factor would be 32 , instead of 2. Considering both the coloring and matching process, the approximate ratio of p-stable algorithm is ( 32 )Δ . Sj ,

7 Performance evaluation In this section, we make evaluation tests on the proposed algorithm “Rematch.” We briefly introduce our configuration parameters, and present the simulation results. Moreover, discussions are made for exploring the proposed algorithm in detail. 7.1 Simulation configuration Simulation parameters are listed in Fig. 1. Our simulation environment is a network with many nodes randomly deployed in a region. Each node is assigned a random number of available channels. Radio interfaces are randomly selected from [1, Rmax ]. The number of channels is randomly selected from [1, Cmax ]. If two nodes are closer, there would be more available channels and interfaces. And each node selects the transmitting range according to the power level, which are listed in Table 1. We compare our algorithm with the pure average value based one-stage matching algorithm. In this paper, we convert the extended graph into confliction graph separately, and use the heuristic algorithm based on the coloring problem [10]. Simulations are done independently 50 times on each scenario, and then we compute the average result among them. Table 1 Simulation parameters

Parameters

Value

Average number of neighbors per node

4, 8, 12, 16, 32

Rmax

4

Cmax

10

Transmitting range (meters)

50, 100, 150, 200

Channel quality values

0.4, 0.5, 0.6, 0.7, 0.8, 0.9

Probability values

0.1, 0.1, 0.1, 0.2, 0.5

Number of flows

5

Transmission rate

10 Mbps


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7.2 Simulation results As shown in Fig. 9, with an increasing number of neighboring nodes, the throughput achieved by one-stage matching drops seriously, while the two-stage matching algorithm maintains a relatively high throughput. The end-to-end throughput in Fig. 9 is of the order kbps. The reason lies in the two-stage scheduling because in a heavy contended network, the channel quality variation would seriously affect network performance. The two-stage matching algorithm could reasonably use the schedule among different contended links and improve network performance. In order to evaluate the performance variations in case of channel dynamics, we intentionally changed our channel distribution listed in Table 1, and the probability distribution has a larger variance. And the values are 0.1, 0.15, 0.25, 0.3, 0.8, 0.9, and the probabilities are listed as 0.1, 0.3, 0.2.0.1, 0.3. We intend to know the throughput variations as the channel states are changing, either on the probability distribution or the transition probabilities. As shown in Fig. 10 with anincreasing number of neighFig. 9 Throughput with increasing no. of neighbors

Fig. 10 Stability of two-stage matching with increasing no. of avg. neighbors

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Fig. 11 Stability of two-stage matching with increasing no. of avg. neighbors

boring nodes, two-stage matching still maintains a relative high throughput with a relatively high stability factor. Our two stage scheduling algorithm effectively utilizes network resources in a highly dynamic environment and maintains a relatively high stability. Simulation results in Figs. 9 and 10 show that ours is a stable and efficient algorithm in an unstable heterogeneous MCMR WMN. In Fig. 11, we change the number of flows in network. And it shows that with the increasing number of flows, the two-stage matching algorithm outperforms one stage matching algorithm. The improvements on the average e2e throughput improve also with the increasing number of flows. We could infer from this phenomena that “Rematch” is applicable in a heavy traffic network environment.

8 Conclusions and future work We make an investigation on a scheduling algorithm in heterogeneous MCMR WMN, which is suffering from the stochastic characteristics of wireless communications. In a highly dynamic environment, achieving the optimal value on a stochastic programming model, is what we have proved in this paper is N P-hard. The two-stage matching algorithm, although is only a heuristic method, can effectively utilize network resources with stochastic channel states. Simulation results show that, in a highly dynamic environment, the two-stage algorithm could achieve a relatively high network performance. More importantly, ours can be effectively working in a distributed network environment, and it is more applicable for real network implementation. Future research works include a better heuristic algorithm and a really deployed WMN implementation to verify our simulation results. Moreover, the channel quality values should be available from the real WMN deployment. Acknowledgements This work is supported in part by the National Basic Research Program of China (973 Program) under grant No. 2006CB303004, No. 2009CB3020402, China 863 project Grant No. 2008AA01Z216, and Jiangsu High-Tech Research Project of China under Grant No. BG2007039.


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P. Yang, G. Chen Panlong Yang received his B.Sc., M.Sc., and Ph.D. degrees in Communication and Information Systems from Nanjing Institute of Communication Engineering, China, in 1999, 2002, and 2005, respectively. During November 2006 to March 2009, he was Postdoc Fellow in the Department of Computer Science, Nanjing University. Doctor Yang is now Assistant Professor in the Nanjing Institute of Communication Engineering. His research interests include wireless mesh networks, wireless sensor networks and cognitive radio networks. Doctor Yang has published more than 30 papers in peer-reviewed journals and refereed conference proceedings in the areas of mobile ad hoc network, wireless mesh networks and wireless sensor networks. He has also served as a member of program committees for several international conferences. He is a member of the IEEE Computer Society.

Guihai Chen obtained his B.Sc. degree from Nanjing University, M. Engineering from Southeast University, and Ph.D. from University of Hong Kong. He visited Kyushu Institute of Technology, Japan in 1998 as a research fellow, and University of Queensland, Australia in 2000 as a visiting professor. During September 2001 to August 2003, he was a visiting professor in Wayne State University. He is now Full Professor and Deputy Chair of the Department of Computer Science, Nanjing University. Professor Chen has published more than 180 papers in peer-reviewed journals and refereed conference proceedings in the areas of wireless sensor networks, high-performance computer architecture, peer-to-peer computing, and performance evaluation. He has also served on technical program committees of numerous international conferences. He is a member of the IEEE Computer Society.