Virtual access network embedding in wireless mesh networks

Ad Hoc Networks 10 (2012) 1362–1378

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Ad Hoc Networks journal homepage: www.elsevier.com/locate/adhoc

Virtual access network embedding in wireless mesh networks Pin Lv a, Xudong Wang b,⇑, Ming Xu a a b

School of Computer, National University of Defense Technology, Changsha, China University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, China

a r t i c l e

i n f o

Article history: Received 26 December 2011 Received in revised form 23 March 2012 Accepted 26 March 2012 Available online 28 April 2012 Keywords: Network virtualization VANE Virtual access network Wireless mesh networks

a b s t r a c t Network virtualization of a wireless mesh network (WMN) is an economical way for different subscribers to customize their exclusive access networks through a common network infrastructure. The most critical task of network virtualization is virtual network embedding, which can be divided into two sub-problems: node mapping and link mapping. Although there exist approaches to virtual network embedding in wired networks, the characteristics of WMNs make virtual network embedding become a unique and challenging problem. In this paper, virtual access network embedding is studied for WMNs. To support flexible resource allocation in virtual access network embedding, each access node is designed based on orthogonal frequency division multiple access (OFDMA) dual-radio architecture. Through subcarrier allocation on each link, virtual access networks are gracefully separated from each other. To coordinate channel assignment across different links under the constraint of a limited number of orthogonal channels, a novel channel allocation algorithm is proposed to exploit partially-overlapped channels to improve resource utilization. Since the virtual access network embedding problem is NP-hard, a heuristic algorithm is developed based on an enhanced genetic algorithm to obtain an approximate but effective solution. Simulation results illustrate that the virtual access network embedding framework developed in this paper works effectively in WMNs. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Wireless mesh network (WMN) [1] is a promising technology to provide wireless clients with high bandwidth and expanded coverage to access the Internet. In WMN, mesh routers establish a multi-hop backbone through wireless links. A mesh router is referred to as a mesh access point (MAP) if it provides accessing service for wireless clients. A mesh router that is connected to the wired Internet is referred to as an Internet gateway (IGW). Multiple IGWs can be deployed in a WMN to increase the network capacity. A wireless client (e.g., laptop, PDA, WiFi IP phone, wireless camera, etc.) can access the Internet after being associated with a MAP. The packets of the client are forwarded by the wireless mulit-hop backbone between the ⇑ Corresponding author. Tel.: +86 021 34207221. E-mail address: [email protected] (X. Wang). 1570-8705/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.adhoc.2012.03.016

MAP and an IGW. As a convenient and low-cost approach to provide Internet access, WMNs will become increasingly widespread in the near future. In practical applications, it is common that different clients have various end-to-end bandwidth requirements when they are sharing a WMN. However, traditional resource allocation, scheduling, or QoS provision schemes in WMNs cannot support diverse users’ requirements. For example, previous resource allocation or scheduling methods [2] aim at entire network performance optimization and can only offer best effort services to clients. The QoS guarantee mechanisms [3] are mainly designed for distinct pre-specified services (such as real-time or multimedia applications) rather than specific clients. Although deploying a private WMN for each client can meet the demand of this client, it has high cost and low flexibility. Network virtualization (NV) of a WMN is an economical and elastic way for service subscribers to customize their

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exclusive access networks through a shared WMN infrastructure. Such exclusive access networks work as virtual access networks (VANs) over the same WMN infrastructure to satisfy diverse end-to-end bandwidth or security requirements of the subscribers. The shared infrastructure is referred to as the physical network or the substrate network. Network virtualization allows a physical network to support multiple virtual networks simultaneously. As demonstrated in Fig. 1, three clients (Clients 1, 2 and 3) have distinct bandwidth demands for Internet accessing and three VANs (VANs 1, 2 and 3) are established for each client. Each virtual access network works independently without any interference with each other. Consequently, each client gets an impression that it is offered with a dedicated access network that satisfies its end-to-end bandwidth requirement. A major challenge of network virtualization is the virtual network embedding (VNE) problem that handles efficient mapping of virtual nodes and virtual links onto physical nodes and links [4,5]. Virtual network embedding is accomplished if two tasks are fulfilled: node mapping and link mapping. In [4,6], the two tasks were completed in two stages, while nodes and links were mapped during the same stage in [7,8]. To solve the virtual network embedding problem, previous research work [4,5,7] was mainly focused on wired networks. However, the characteristics of a WMN make virtual access network embedding (VANE) a unique and challenging research problem. Unlike those existing solutions in wired networks, the node mapping of VANE in a WMN is only partially deterministic, as the MAP of a VAN request is simply selected according to the location of VAN users, but the IGW of the VAN has to be determined based on throughput optimization. Due to the broadcast nature of wireless links, the link mapping of VANE needs to consider the specific

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multiple access mechanism of a WMN, and then it determines resource allocation for each VAN accordingly to guarantee the independence of each VAN. In this paper, orthogonal frequency division multiple access (OFDMA) is considered as the multiple access scheme to ensure fine-grained and flexible resource allocation. Through subcarrier allocation on each link, virtual access networks are gracefully separated from each other. To coordinate channel assignment across different links under the constraint of a limited number of orthogonal channels, a novel channel allocation algorithm is proposed to exploit both non-overlapping and partially-overlapped channels to improve the resource utilization. If the total bandwidth requirement exceeds the capacity a WMN, a subset of the virtual access networks need to be selected and embedded into the WMN. Furthermore, network resources must be utilized efficiently to maximize the revenue of the network provider. Since the virtual access network embedding problem is NP-hard, a heuristic algorithm is developed based on an enhanced genetic algorithm to obtain an approximate solution. Simulation results based on the heuristic algorithm illustrate that the virtual access network embedding framework developed in this paper works efficiently and effectively in WMNs. To the best of our knowledge, this is the first effort to solve the virtual access network embedding problem in a WMN environment and our contributions are listed as follows:  An OFDMA-based mesh node and WMN architecture are designed to support fine-grained and flexible resource allocation for virtual access network embedding.  A novel channel allocation algorithm is proposed, which exploits both non-overlapping channels and

Fig. 1. Virtual access networks in wireless mesh network.

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partially-overlapped channels to increase the utilization of WMNs.  A heuristic algorithm is developed based on the formulation of optimal virtual access network embedding through an enhanced genetic algorithm. It effectively accomplishes virtual access network embedding in WMNs. The remainder of the paper proceeds as follows. In Section 2, the related work is summarized, while in Section 3 the challenges and possible approaches to virtual access network embedding are discussed. In Section 4, an OFDMA-based WMN architecture is designed to support virtual access network embedding. A novel channel assignment algorithm under this architecture is proposed in Section 5. The virtual access network embedding problem is formulated in Section 6. In Section 7, an enhanced genetic algorithm is developed based on a greedy algorithm and a genetic algorithm to obtain an approximate solution to the virtual access network embedding problem. Simulation results are presented in Section 8 and the paper is concluded in the last section.

2. Related work Network virtualization (NV) has been considered as a powerful scheme to boost the evolution of the future Internet by allowing heterogeneous architectures and different protocols to coexist over a shared physical infrastructure [9]. It has attracted great attentions from both academic and industrial communities in recent years. Surveys on network virtualization are provided in [10,11], where techniques such as virtual local area network (VLAN), virtual private network (VPN), active and programmable networks, overlay networks are summarized. The most significant past and on-going projects related to network virtualization as well as research challenges are also listed in their papers. Virtual network embedding (VNE) is one of the most challenging problem in network virtualization. Many prior works [4,6,12] divide the VNE problem into two independent stages: node mapping and link mapping. The node mapping usually uses greedy strategy, while the link mapping is mainly based on shortest path algorithms. Online schemes are designed in [6] to achieve low and balanced load on both nodes and links in substrate networks. Lu et al. [12] propose an off-line scheme for specific virtual topology. A VNE mechanism which allows flow split and path migration is proposed in [4]. Since backtracking is usually inevitable during the two-phase implementation of VNE, it leads to low efficiency in searching the optimal solution. In order to achieve better performance, VNE algorithms are proposed in [7] to better coordinate the two stages. In [13], a distributed algorithm is designed to simultaneously map virtual nodes and virtual links without any centralized controller. The authors in [8] adopt a method called subgraph isomorphism detection to match the topology of virtual network in the substrate network. Furthermore, research in [5] aims at solving the VNE problem in evolving networks, and the survivable problem of VNE is studied in [14].

All the above-mentioned research work is focused on the environment of wired networks. In the field of wireless networks, only a little related work can be found. Due to the broadcast nature of wireless links, link virtualization of VNE is more challenging in wireless networks. Based on the characteristics of wireless environment, a report from the GINI project [15] describes ‘‘virtualization’’ and ‘‘slicing’’ of wireless networks to enable concurrent operation of multiple experiments on a common wireless network. Park et al. propose a general framework for VNE in wireless networks [16]. Matos et al. analyze the model of context-based multivirtual wireless mesh networks [17], and discuss a case of context-based wireless mesh network virtualization in [18]. Moreover, the technique for wireless mesh network slicing to accommodate several experiments simultaneously is presented in [19]. However, none of them takes specific virtual access network demands into consideration.

3. Challenges of VANE in WMNs Due to the broadcast nature, a wireless link is a shared transmission medium. Consequently, the key challenge in wireless network virtualization is to virtualize wireless links [20]. A fundamental principle of virtual access network embedding (VANE) is that, when multiple virtual access networks (VANs) co-exist in the same physical WMN, the activities of one VAN should not affect any other VANs, and vice versa. This principle can be further expanded into the following two rules [20]:  Coherence. When a transmitter of a VAN is active, all of its corresponding receivers and potential sources of interference as defined by the VAN should be simultaneously active on their appropriate channels of operation.  Isolation. When a node of one VAN is receiving packets pertinent to the VAN, no transmitter of a different VAN within the communication range of the receiver should cause co-channel or inter-channel interference. In order to meet these two requirements, wireless resources should be carefully assigned to VANs. VANs should be separated from each other in at least one of the domains including space, frequency, time, and code. Hence, corresponding approaches such as space division multiple access (SDMA), time division multiple access (TDMA), frequency division multiple access (FDMA), code division multiple access (CDMA) or hybrid approaches can be adopted [15,20]. However, these approaches cannot solve the VANE problem in WMNs directly. More specifically, the space of VANs is not totally determined by service subscribers since the subscribers need to select their nearby MAPs. Thus, the SDMA approach is not applicable for the scenario discussed in this paper. The TDMA approach is not flexible since the overhead incurred by guard time is heavy when the time domain is divided into fine-grained slots. In the traditional FDMA and CDMA methods, the number of virtual nodes that can be mapped to a substrate node is determined by how many radios the substrate node has. This will lead to poor flexibility and scalability.

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Throughput (Mbps)

In next section, we design an OFDMA-based WMN architecture which supports VANE in a fine-grained level of wireless resources. 4. OFDMA-based WMN architecture design for VANE 4.1. Experimental observation: out-of-band interference effect

4.2. OFDMA-based mesh node and architecture design The OOB interference results in remarkable performance degradation. Raman et al. [22] designed a new MAC protocol to avoid this problem. In their work, the transmission and reception behaviors of the radios on the same node are synchronized to make sure all the radios transmit or receive packets at the same time. However, their MAC protocol has to disable acknowledgement (ACK) to avoid unexpected transmission. As the OOB interference effect among channels in different bands can be ignored, a mesh node can be equipped with two radios operating on IEEE 802.11b/g (2.4 GHz) and IEEE 802.11a (5 GHz) channels, respectively. Since those two radios are used to establish multi-radio multi-channel WMN backbone, more radios should be involved to provide access service for clients if the mesh node serves as a MAP. In order to eliminate the influence of OOB interference, the access point radio should be installed on another board

Fig. 2. The OOB interference experiment setup.

12 9 6 3 0 1 2 3 4 5 6 7 8 9 10 11

Link 1

12

34

7 56

11 10 89

Link 2

Fig. 3. End-to-end throughput of channel assignment in 2.4 GHz band.

30 25

Throughput (Mbps)

Multi-radio and multi-channel technology helps us to improve the performance of a WMN. When a wireless node is equipped with multiple radios, the out-of-band (OOB) interference effect usually exists, which is caused by imperfect filtering or phase/LO (local oscillators) noise of the radios [21]. The experiment shown in Fig. 2 is carried out to display the OOB interference effect. Node C, who owns two radios, receives A’s packets from one interface and then forwards them to B from the other interface. Both of the data links (Link 1 and Link 2) vary their channels from 1 to 11, and the end-to-end throughputs under different channel assignments are tested. According to the results in Fig. 3, the end-to-end throughputs from A to B have no significant differences when we alter the channel assignments, even the two links are assigned two non-overlapping channels. If two links are configured with channels in different bands (one in 2.4 GHz and the other in 5 GHz), the throughput is dramatically higher as shown in Fig. 4. It can be concluded that assigning radios of one node with non-overlapping channels in the same frequency band cannot avoid interference due to the OOB emission. However, when the radios operate on channels belonging to different bands, the OOB effect is weak enough to avoid mutual interference.

15

both in 2.4GHz band one in 2.4GHz one in 5GHz band

20 15 10 5 0

Different channel assignment scheme Fig. 4. End-to-end throughput comparison of channel assignment in different bands.

and the board is connected to the dual-radio mesh node through ethernet cables and a hub, as shown in Fig. 5. Many wireless interface drivers, such as Madwifi, support multiple virtual access points (VAPs) by one radio. Each VAP has its own ESSID and encryption scheme. Therefore, in terms of an access point, we create VAPs for each embedded VAN request. In terms of a WMN backbone, we use orthogonal frequency division multiple access (OFDMA) as the multiple access scheme. Distinct virtual access networks (VANs) are assigned with different subcarriers to be gracefully separated. If the bandwidth of a single subcarrier is b0 and the bandwidth requirement of a VAN is Bi, the number of the subcarriers n that should be allocated to the VAN is:

n¼

  Bi b0

ð1Þ

When a VAN requests to transmit but another VAN on the same physical node is receiving data, the transmitter will not be able to transmit until the receiver finishes its operation. To avoid such a contradictory case, we require one radio is used for transmission (called Tx radio) and the other radio is for reception (called Rx radio). The mesh node architecture is illustrated in Fig. 6. To avoid violating ACK scheme, the ACK transmission is viewed as atomic operation together with data reception, and vice versa. More specifically, after transmitting data packets through different subcarriers, the Tx radio will wait for ACKs. Similarly, when the Rx radio has received data packets successfully, it will send ACKs after an SIFS (short interframe spacing) for Rx/Tx switching. The Tx fragmentation layer is used to fragment data packets of different VANs into blocks of same length, while the Rx reassembling layer

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Fig. 5. Mesh access point node.

Fig. 6. OFDMA-based mesh node design for VANE.

is used to rebuild the packets. In this way, each VAN can start transmission immediately without waiting for other VANs’ receptions. Based on the mesh node design, the WMN backbone is constituted as depicted in Fig. 7. All the mesh nodes routed to the same Internet gateway (IGW) and their routing paths form a tree structure rooted at the IGW, which is referred to as IGW-tree. If the IGW is more powerful to be equipped with more radios, it can be considered as multiple IGWs. To guarantee the connectivity, the nodes in the same IGWtree operate on two channels in different bands: one for transmission, the other for receiving. This mechanism does not cause performance degradation since the VANs converged to the same IGW are mapped to distinct subcarriers. The interference within one VAN, which is caused by the multi-hop nature of the WMN, is also minimized. Different IGW-trees should be assigned with different channels to eliminate mutual interference. In the next section, a channel assignment algorithm for the trees will be proposed. 5. Channel assignment algorithm for IGW-trees Fig. 7. OFDMA-based WMN architecture for VANE.

5.1. Experimental observation: partially overlapped channels The IEEE 802.11b/g standards specifications define 11 channels (for the FCC domain) or 13 channels (for the ETSI

domain) in 2.4 GHz ISM band. We consider the 11 channels in the FCC domain which are shown in Fig. 8. The bandwidth of each channel is 22 MHz, whereas the center

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5.2. Channel assignment algorithm for IGW-trees Based on the architecture design, there are multiple IGW-trees in the network, one for each IGW. Different trees should be assigned with various channels to avoid

Fig. 9. The POC experiment setup.

45

0 1

40

Throughput (Mbps)

frequency interval of adjacent channel pair is only 5 MHz. This causes that every channel is partially overlapped with several channels relatively close to it. Two channels separated by at least 25 MHz are referred to as ‘‘nonoverlapping’’ (orthogonal). The maximum number of nonoverlapping channels can be used is three (namely the Channels 1, 6 and 11). The majority of traditional channel assignment schemes (e.g., [23,24], et al.) are based on the non-overlapping channels (NOCs). Due to the limited number of such channels, it is inevitable to assign identical channel to links in physical proximity, especially when the wireless nodes and links are of high density. In the case of VANE in WMNs, the VAN requests associated with different IGWs must be assigned with proper channels to avoid mutual interference. Recent results [25–27] have revealed that the capacity of wireless networks can be further increased by utilizing both non-overlapping and partially-overlapped channels (POCs) with careful design. In order to quantify the degree of the interference between various partially overlapped channels, we measure their influence on the UDP throughput using the experiment setup depicted in Fig. 9. Two pairs of wireless nodes with commodity 802.11 hardware (Lenovo Thinkpad laptops with 802.11 a/b/g wireless interfaces) are located in a relatively empty playground. The two nodes in each pair are placed in close proximity to each other and UDP traffic is sent between them. The transmission power of the nodes is set to 1 mW (0 dBm) which is the lowest level supported by the interfaces. The total throughput of the two links versus physical distances and channel distances is shown in Fig. 10. The curves in Fig. 10 indicate the fact that the effect of ‘‘zero-interference’’ can be achieved at much lower physical distance by increasing the channel separation between the two links. For instance, as shown in Fig. 10, a channel distance of two (say Channels 1 and 3) is enough for both links to transmit without interference with a physical distance of about 50 m, while the channel separation can be set to three when the physical distance is 40 m. However, if being configured on the same channel, two links should be separated by more than 70 m to avoid interference. It follows that the partially-overlapped channels can enhance spatial reuse if used carefully.

2 3

4 5

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0

10

20

30

40

50

60

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Physical Distance (m) Fig. 10. Total throughput over different physical and channel distance.

interference. Due to the limited non-overlapping channels (especially the radio in 2.4 GHz band), a channel assignment scheme exploiting both non-overlapping channels and partially-overlapped channels is developed for IGW-trees. The WMN to be assigned with channels for different virtual networks can be depicted as a complete graph Gd with each node in the graph representing an IGW-tree in the WMN. The weight of an edge between two nodes signifies the physical distance between the corresponding IGWtrees. The distance between two IGW-trees is defined to be the minimum distance between nodes from each tree. Suppose the critical distance of channel separation has been measured. Then the graph Gd can be transformed into the conflict graph Gc in which the edge weight indicates the minimal channel separation between the two corresponding IGW-trees. The channel assignment problem can be described as: Given a channel set C and a conflict graph G = (V, E) where jVj = n, find a channel vector hc1, c2, . . . , cni to satisfy that, for each eij 2 E, jci  cjj P w(eij), where ci 2 C

Fig. 8. Available channels in IEEE 802.11b/g frequency band.

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Fig. 11. An illustration of channel assignment process.

(i = 1, . . . , n) and w(eij) is the least channel separation between node i and node j. The channel assignment problem is proved to be NPcomplete, as shown in Section A. We propose a polynomial-time approximation algorithm as listed in Algorithm 1. The complexity of the algorithm is O(jVj2). Algorithm 1 initializes two sets firstly. The set S includes the node which has been assigned a channel, while the set R contains the nodes that have not been assigned any channel yet. The main idea of the Algorithm 1 is to assign a channel to the node with the largest degree in R first. The reason why the largest-degree node has the highest priority is that the largest-degree node has the most constrains of the neighbors. Once the largest-degree node has been assigned the channel ci, its neighbor j can obtain a candidate channel set C⁄ such that, for each cj 2 C⁄,

jci  cjj P w(eij) holds. When assigning a channel to a node, the channel is selected from the intersection of candidate channel sets of every round. Any channel in the intersection of the candidate channel sets satisfies the interference constrains. In our channel assignment algorithm, if there are multiple candidate channels, the channel with the minimal sequence number is selected. Once a node (e.g., v) has been assigned a channel, it is added into the set S. In the next round, the set of nodes that can be assigned channels is determined in two cases: if the neighbors of v have not been allocated a channel, they form the set R; otherwise, R consists of all nodes excluding v, i.e., the set V  S. After each node has been assigned a channel, the algorithm terminates with success. Once the intersection equals an empty set and the node has no available channels to assign, the algorithm stops and reports failure.

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Algorithm 1. Channel Assignment (Gc(V, E), C) Input: Gc(V, E): a conflict graph; C: available channel set. Output: Channel assignment results. 1: S ; 2: R VS 3: for vi 2 V do 4: Ci C 5: end for 6: while S – V do 7: Select vi 2 R with the largest degree 8: Get the assignable channel set C⁄ of vi 9: Ci Ci \ C⁄ 10: if Ci – ; then 11: Assign c 2 Ci to vi, where c is the minimalsequence-number channel in Ci 12: S S [ {vi} 13: else 14: Stop, return false 15: end if 16: if vi has unassigned neighbors then 17: R {vi’s unassigned neighbors} 18: else 19: R VS 20: end if 21: end while 22: Stop, return true. The channel assignment process of Algorithm 1 is illustrated in Fig. 11. The original conflict graph is shown in Fig. 11(1). Currently, the set S is an empty set, while R is equal to the node set V. We use notation [ci, cj] to denote the set of channels from ci to cj which are candidate channels of a node. In the initial phase, candidate channel set of each node is [1, 11]. One of nodes with the largest degree, i.e. node a, is assigned Channel 1 in the first round, as shown in Fig. 11(2). The assigned node as well as its channel is marked by the grey. After node a has been assigned a channel, it is added into S, and R becomes the set of a’s neighbors which have not been assigned any channel. Meanwhile, the candidate channel sets of all the nodes in R are updated based on weights of edges. Next, the largest-degree node in R is assigned a channel based on its candidate channel set, and the process goes on in the same manner. After node e has been assigned a channel (in Fig. 11(5)), since all its neighbors have been assigned channels, R is set to V  S which equals {c}. When the set S is equal to V, the channel assignment algorithm terminates with success (in Fig. 11(6)). With suitable channel assignment, nodes in different IGW-trees do not have to contend for transmission media. The VANs associated with the same IGW-tree are separated through subcarrier allocation. In this way, a VAN operates independently without interfering each other. 6. Problem formulation of VANE 6.1. Model of physical wireless mesh network We consider a deployed WMN denoted by a weighted directed graph ~ G ¼ ðN; ~ EÞ, where N refers to the set of mesh

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nodes (including MAPs and IGWs) and ~ E refers to the set of possible directed links in the WMN respectively. Each mesh node n 2 N is associated with a location loc(n). We use I to indicate the set of IGWs and A to indicate the set of MAPs, where I  N, A  N and I \ A = ;. The number of the IGWs in the WMN is m and the number of MAPs is k, so that jIj = m, jAj = k. Each possible link ~ eij 2 ~ E from mesh nodes i to its neighbor j is associated with the available bandwidth capacity bð~ eij Þ which can be measured using rate control schemes in actual environments. Based on the mesh node design for VANE, the weighted directed graph ~ G can be transformed into a weighted undirected graph G = (N, E). For each eij 2 E, bðeij Þ ¼ minðbð~ eij Þ; bð~ eji ÞÞ, where ~ eij 2 ~ E; ~ eji 2 ~ E. The capacity of IGW i is defined to be the maximum link capacity with its neighbors:

C i ¼ maxbðeij Þ eij 2E

ð2Þ

The symbol P is defined to be the set of all the paths in the physical WMN, and each multi-hop path pij 2 P is constituted by successive links from node i to node j without loops. A path pij = (Np, Ep) is a subgraph of G. The available bandwidth of pij is determined by the link with the minimum bandwidth in this path:

bðpij Þ ¼ min BðeÞ p e2E

ð3Þ

When being mapped to a virtual link, the physical link needs to be assigned with a wireless channel f(e) 2 F, where F is the set of available channels. The number of available channels and their frequencies are defined by the wireless communication standards used by the mesh nodes. In this paper, IEEE 802.11 standards are considered. Hence, the bands of 2.4 GHz and 5 GHz are available. Assume an MAP forwards packets to only one IGW and multi-path routing is not taken into account. When all the routing paths have been established, there will be m IGWtrees rooted at the IGWs. The physical distance between two substrate nodes i and j is marked as D(i, j). The physical distance between two IGW-trees (e.g., Tp and Tq) is defined to be the minimum distance between the node of each tree:

DðT p ; T q Þ ¼ min Dði; jÞ i2T p ;j2T q

ð4Þ

If two trees do not interfere with each other under certain channel assignment ~ f , it can be denoted as

I ðT p ; T q ; ~ fÞ ¼ 0

ð5Þ

The traffic load carried by the IGW-tree Tp is denoted as L(Tp) which should not exceed the capacity of the IGW. 6.2. Model of virtual access network request A virtual access network (VAN) request can be modelled as a weighted directed graph ~ GV ¼ ðN V ; ~ EV Þ, as shown in Fig. 12. The node set NV contains two virtual nodes: a virtual MAP and a virtual IGW. The location of the virtual MAP is deterministic since the client decides which MAP it can be connected to according to its location. Hence, the virtual

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MAP is simply mapped to the physical MAP with the same location. The physical IGW that the virtual IGW is mapped to is to be determined when conducting VANE. The link set ~ EV contains two virtual links. The virtual link from the MAP to the IGW is referred to as the up-link while the reverse virtual link is defined to be the downlink. Each virtual link ~ e 2~ EV is associated with a bandwidth request Bð~ eÞ. The bandwidth requests of up-link and downlink are allowed to be different. For example, if a client wants to access the Internet to watch movie, the down-link traffic is dominated; if a client is a wireless camera which is used for remote monitoring, the up-link traffic is higher. The weighted directed graph ~ GV can be transformed into a weighted undirected graph GV = (NV, EV). For each ~ ~ eij 2 EV ; Bðeij Þ ¼ Bð~ eij Þ þ Bð~ eji Þ, where ~ eij 2 EV ; ~ eji 2 EV . 6.3. Problem formulation A WMN (namely G) and a set of VAN requests (represented by V) are given. Due to the capacity limitation of the WMN, if not all VAN requests can be supported, the problem of virtual access network embedding (VANE) in WMN is to find a subset of VAN requests VI # V to be embedded into the WMN with the objective of revenue maximization. We assume the expense that a service subscriber pays to the infrastructure provider is proportional to the total end-to-end bandwidth it requests. Therefore, maximizing the revenue of the infrastructure provider is equivalent to maximizing the bandwidth utilization of the WMN. To embed multiple VANs onto a substrate WMN, three tasks need to be fulfilled:

above-mentioned three tasks interact with and restrict each other, our proposed VANE problem is even more complicated and challenging than the MKP. Thus, the VANE problem is also NP-hard, and its optimal solution cannot be obtained in polynomial time. The VANE problem can be formulated as: Given a weighted undirected graph G = (N, E) where A  N, I  N, jAj = k, jIj = m; a set V where jVj ¼ n and for each j 2 V, there is Bj > 0; a mapping M : V ! A; and a set F, we need to find the following three vectors:  an n-dimensional vector ~ x ¼ hx1 ; x2 ; . . . ; xn i, where xi 2 {0, 1}, i = 1, . . ., n. If xi = 1, it means the ith VAN is supported by the WMN; otherwise, it cannot be embedded when xi = 0.  an m-dimensional vector ~ T ¼ hT 1 ; T 2 ; . . . ; T m i, where the notation T i ¼ ðN T i ; ET i Þ indicates the IGW-tree rooted at the ith IGW, N T i # N; ET i # E; i ¼ 1; . . . ; m. It is satisfied that 8i; j ¼ 1; . . . ; m; i – j () NT i \ NT j ¼ ;; ET i \ ET j ¼ ;.  an m-dimensional vector ~ f ¼ hf1 ; f2 ; . . . ; fm i, where fi 2 F, i = 1, . . ., m. The notation fi represents the ith IGW-tree is assigned channel fi. The objective is to

maximize

n X Bi xi ;

ð6Þ

i¼1

X

subject to

Bj 6 ci ; i 2 I; j 2 V

ð7Þ

MðjÞ2N T i

bðPik Þxj P Bj xj ; i 2 I; j 2 V; k ¼ MðjÞ; k 2 NT i ð8Þ

(1) VAN Request Selection. Since the WMN cannot support all the VAN requests owing to the capacity limitation of the WMN, a subset of the VAN requests should be selected to be embedded into the WMN. (2) IGW-Tree Establishment. After VAN request selection, routings should be established between corresponding MAPs and IGWs, and IGW-trees should be built. (3) Channel Assignment. To keep the isolation of VANs, each IGW-tree should not interfere the others through proper channel assignment. We use Algorithm 1 to increase the resource utilization in the premise of guaranteeing isolation between VANs.

f Þ ¼ 0; i 2 I; l 2 I; i – l I ðT i ; T l ;~ xj 2 f0; 1g; j 2 V

maxBj 6 maxci

ð11Þ

minBj 6 minci

ð12Þ

n X Bj > maxci

ð13Þ

j2V

j¼1

Fig. 12. A VAN request.

ð10Þ

Inequality (7) ensures that total bandwidth requirement of VANs associated with an IGW should not exceed the capacity of the IGW. Inequality (8) guarantees the bandwidth of the routing path of a VAN satisfy the bandwidth request of the VAN. Eq. (9) implies that there is no interference between any two IGW-trees under the channel assignment ~ f . Constraint (10) means xj = 1 if the jth VAN is embedded and xj = 0 otherwise. It should be noted that the coefficients Bj and ci are positive integers. Additionally, we have

j2V

If only VAN request selection is considered, the VANE problem will be simplified into a well-known NP-hard 0–1 Multiple Knapsack Problem (MKP). Since the

ð9Þ

i2I

i2I

i2I

The assumption (11) ensures that the traffic of each VAN request can be forwarded to at least one IGW; otherwise, it may be removed from the problem. If the inequality (12) is violated, the IGW with the lowest capacity can be ignored, as it supports no VAN requests. The assumption (13) avoids a trivial solution where all VAN requests are supported by the highest-capacity IGW.

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1371

7. VANE algorithms

7.2. Genetic algorithm

Since the virtual access network embedding (VANE) problem is too complicated, the IGW-tree establishment process needs to be simplified. When the IGW is determined, only the shortest path from an MAP to the IGW is selected. It can be computed through Dijkstra Algorithm. P The metric of path p used in this paper is ei 2p r1i where ri is the bit-rate of the link ei. Even the IGW-tree establishment process has been simplified, the VANE problem is still NP-hard, and the optimal solution cannot be obtained in polynomial time.

Heuristic algorithm is another method to efficiently search optimal solutions of NP-hard problems. We also develop a heuristic, more specifically, a genetic algorithm (GA) as listed in Algorithm 3, to solve the VANE problem. The concept of a chromosome in a genetic algorithm points to an individual solution. In our algorithm, the chromosome is an n-dimensional vector ~ x with n genes:

7.1. Greedy algorithm A greedy algorithm is developed as listed in Algorithm 2. In the greedy algorithm, each MAP selects the nearest IGW to establish its routing path. Based on the optimality principle [28], the routing path from one MAP to its nearest IGW will not go through MAPs in another IGW-tree. Thus, the greedy algorithm ensures that the IGW-trees are separated. The greedy algorithm first tries to embedded the virtual access network (VAN) with the largest bandwidth requirement. Once the VAN is selected, the MAP of the VAN and the IGW of the MAP are determined. The VAN can be embedded if the following three conditions are met: (1) residual capacity of the IGW satisfies the bandwidth requirement of the VAN; (2) the bandwidth of substrate path satisfies the bandwidth requirement of the VAN; (3) channels are assigned successfully when the MAP is routed to the IGW. After all the VANs are checked to be embedded, the greedy algorithm terminates. Since the complexity of Algorithm 2 is OðjAjjIj þ jVjÞ, the greedy algorithm can obtain an approximate solution to the VANE problem quickly. Algorithm 2. VANE-Greedy 1:for each a 2 A do 2: Compute the shortest paths from each IGW to a 3: Choose the IGW with the lowest routing metric as the IGW of a 4: end for 5: for v i 2 V with the largest bandwidth request do 6: Get the MAP aj of vi and the IGW gk of aj 7: if C(gk) P B(vi) & & b(p(aj, gk)) P B(vi) & & channel assignment is successful when aj is routed to gk then 8: xi = 1 9: Update C(gk) and b(p(aj, gk)) 10: else 11: xi = 0 12: end if 13: V V  fv i g 14: end for 15: Stop, return the embedding solution.

~ x ¼ hx1 ; x2 ; . . . ; xn i; where xi 2 {0, 1} and i = 1, 2, . . ., n. The gene xi corresponds to the ith VAN request. The ith VAN is embedded into the WMN if xi = 1. If xi = 0, it means the ith VAN request cannot be accepted by the WMN. The fitness value of a chromosome is defined to be the total end-to-end bandwidth of all the accepted VANs represented by the chromosome. Algorithm 3. VANE-Genetic 1: Generate the initial population of chromosomes 2: Generations 1 3: while Generations 6 predefined-generation do 4: for each chromosome in the population do 5: Generate m separated IGW-trees randomly 6: while channel assignment is failed do 7: Remove one IGW-tree and the VANs supported by this IGW-tree 8: end while 9: for each IGW do 10: while the total bandwidth of VANs supported by the IGW-tree exceeds the capacity of the IGW do 11: Randomly discard a VAN request 12: end while 13: end for 14: for each VAN being mapped do 15: if the bandwidth request of the VAN exceeds the bandwidth of the substrate path then 16: Discard the VAN request 17: end if 18: end for 19: end for 20: Compute the fitness values of the chromosomes 21: Sort the chromosomes and reserve the best one 22: Crossover other chromosomes 23: Mutate the chromosomes 24: Generations Generations + 1 25: end while 26: Stop, return the best chromosome. In the genetic algorithm, several pieces of chromosomes are preserved to maintain a population. The initial chromosomes are generated in an arbitrary way, e.g., generated randomly or by other algorithms. For each chromosome, m separated IGW-trees are generated randomly. If channel assignment fails based on the

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conflict graph of these IGW-trees, one of these IGW-trees is randomly selected and removed. Accordingly, the VANs mapped to the removed IGW-tree also need to be removed. After each IGW-tree is assigned an appropriate channel, it is needed to check whether Constraints (2) and (3) are violated. If the total bandwidth of all the VANs being mapped to an IGW exceeds the capacity of the IGW, a VAN is randomly selected and discarded. This process repeats until Constraint (2) is met. If the bandwidth of the mapped substrate path is less than the bandwidth requirement of the VAN, the VAN will be discarded to avoid violating Constraint (3). In every generation, the chromosomes are sorted based on their fitness values. The best chromosome directly enters the next generation, and the worst chromosome is obsoleted. This strategy ensures that the new generation will not be worse than the elder generation, and the solutions generated by every generation are monotonically non-decreasing. Other chromosomes are processed through genetic operators (e.g., crossover and mutation) to generate new individuals. The concept of crossover refers to that a pair of ‘‘parent’’ chromosomes exchanges a certain proportion of their genes randomly to produce ‘‘child’’ chromosomes. In the mutation procedure, the genes of a single chromosome are altered with a certain probability to create a new individual. After evolution of a number of generations, the genetic algorithm can find a relatively satisfactory solution. The parameters of crossover probability (PC) and mutation probability (PM) will be determined in Section 8.2. 7.3. Enhanced genetic algorithm In order to accelerate convergence of Algorithm 3, the genetic algorithm can be enhanced with greedy strategies to revise a chromosome if it violates the IGW capacity constraints or utilizes resource inefficiently. (1) IGW Overload: If the total bandwidth of the VANs associated with an IGW exceeds the capacity of the IGW, sort these VANs according to their bandwidth requirements. Remove the VAN with the least bandwidth requirement repeatedly until the IGW is not overloaded. (2) IGW Underload: If there is a gap between an IGW’s load and its capacity, it indicates the resource of the IGW is underutilized. In this case, the unmapped VANs corresponding to the IGW-tree are sorted according to their bandwidth requirements. The VAN with the largest bandwidth requirement is mapped to the IGW repeatedly, until no more VANs can be mapped to the IGW. In addition, in that Algorithm 3 generates solutions randomly, it cannot guarantee the qualities of the solutions. To ensure that the solution obtained by the genetic algorithm is no worse than the solution of the greedy algorithm, Algorithms 2 and 3 are combined to constitute an enhanced genetic algorithm as illustrated in Fig. 13. The solution obtained by the greedy algorithm is used as one of the first generation chromosomes in the genetic

algorithm. Consequently, the final solution will not be worse than the greedy solution after the evolution. 8. Performance evaluation 8.1. Simulation setup We conduct simulations to evaluate our virtual access network embedding (VANE) framework. The topology of the WMN is generated randomly, and locations of mesh nodes follow uniform distribution. The bit-rate of the links depend on the physical distance between two nodes, which is shown in Table 1. The mapping relationship between physical distance and channel distance is listed in Table 2. It specifies the least channel distance of two links when they locate within a certain range. The channel distances in Table 2 do not match the channel distances shown in Fig. 10, because the results in Fig. 10 are obtained when the transmission power of the wireless interface is set to the lowest level (1mW). In our simulation, however, it is assumed that the transmission power of wireless node is at normal level, i.e., dozens of milliwatts. Eight chromosomes are maintained in each generation of the genetic algorithm, and the termination condition is set to evolution for 100 generations. A hundred VAN requests are generated randomly with the bandwidth requirements uniformly distribute from 1Mbps to 20Mbps. Two metrics, i.e., Total Utilized Bandwidth (TUB) and Resource Utilization Ratio (RUR), are measured in the following simulations. The total utilized bandwidth is the sum of the allocated end-to-end bandwidth of all the embedded VANs:

TUB ¼

X Bj xj ;

ð14Þ

j2V

where Bj is the bandwidth requirement of the jth VAN, and xj = 1 if it is embedded, otherwise xj = 0. The resource utilization ratio is calculated by:

P j2V Bj xj RUR ¼ P ; i2I C i

Fig. 13. Framework of enhanced genetic algorithm.

ð15Þ

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P. Lv et al. / Ad Hoc Networks 10 (2012) 1362–1378 Table 1 Bit-rate with respect to distance. 0–50 54 100–150 11

50–100 36 150–200 1

Table 2 Channel distance with respect to physical distance. Physical Channel Physical Channel

distance (m) distance distance (m) distance

0–10 5 80–150 2

10–40 4 150–250 1

40–80 3 250–1 0

Total Utilized Bandwidth (Mbps)

Distance (m) Bit-rate (Mbps) Distance (m) Bit-rate (Mbps)

260 250 240 230 220 210

PM =0.10 PM =0.30 PM =0.50 PM =0.70 PM =0.90 PM =1.00

200 190 180

0

20

40

60

80

100

Generations where Ci is the capacity of the ith IGW.

Fig. 14. Convergence speed with different mutation probabilities.

8.2. Parameter determination

8.3. Simulation results Two scenarios are considered in simulations, which are described as follows: Scenario I: Fixed-Scale WMN. Fifty MAPs are deployed in a 400 m  400 m rectangular area to form a backbone of a WMN. Scenario II: Changing-Scale WMN. The size of an area changes from 300 m to 1000 m while the number of the MAPs varies from 40 to 100. The detailed settings are listed in Table 4. In both scenarios, the number of IGWs (denoted by m) varies from 3 to 8. In the following part, the greedy algorithm (Algorithm 2) is denoted by GR, the genetic algorithm (Algorithm 3) is indicated by GA, and the enhanced

Total Utilized Bandwidth (Mbps)

In the genetic algorithm, two parameters, i.e., the crossover probability (PC) and the mutation probability (PM), need to be determined through simulations. In a 500 m  500 m area, a WMN with 70 MAPs and 8 IGWs is deployed. To evaluate the mutation probability impact on the convergence speed, PM varies from 0.1 to 1 while PC is fixed at 0.5. The convergence speed of the genetic algorithm with different mutation probabilities is revealed in Fig. 14. It can be concluded that the genetic algorithm achieves the fastest convergence speed when PM equals 0.9. Next, the mutation probability is set to 0.9, and the crossover probability impact on the convergence speed is measured by changing the value of PC from 0.1 to 1. The convergence results are shown in Fig. 15. The results indicate that the maximal total utilized bandwidth is reached when PC = 0.3 and PC = 0.9. However, the gap between the maximal and the minimal total utilized bandwidth is only about 4%. It can be concluded that the crossover probability has insignificant effect on the convergence speed when PC is set to different values. Considering the randomness of the genetic algorithm, PC is set to 0.5 in the following simulations. The default parameters in the following simulations are summarized in Table 3.

270 260 250 240 230 220

PC =0.10 PC =0.30 PC =0.50 PC =0.70 PC =0.90 PC =1.00

210 200 190

0

20

40

60

80

100

Generations Fig. 15. Convergence speed with different crossover probabilities.

genetic algorithm is represented by EGA. If an algorithm X exploits partially overlapped channels when assigning channels for IGW-trees, it is marked as X-POC. On the contrary, it is marked as X-NOC if X only utilizes non-overlapping channels. The convergence speeds of GA and EGA are compared when eight MAPs are located in the WMN in both Scenarios I and II, and the results are shown in Fig. 16. With the same channel assignment mechanism, EGA converges faster than GA, and tends to obtain better solutions. It indicates that greedy strategies can accelerate the convergence of GA. Since solutions calculated by GR are initial chromosomes of EGA, the result of EGA will remain flat if EGA does not find a better solution during evolution. The total utilized bandwidths of different algorithms in both scenarios are shown in Fig. 17. The straight line in Fig. 17(a) and Fig. 17(b) indicates the total capacity of all the IGWs. The total capacity, which is the theoretical upper bound of the total utilized bandwidth, increases linearly as more and more IGWs are deployed in the WMN. For each algorithm, the total utilized bandwidth under POC assignment is always higher than the total utilized bandwidth under NOC assignment scheme. Exploiting partially overlapped channels for IGW-tree channel assignment can improve resource utilization. In Scenario I (Fig. 17(a)), the

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total utilized bandwidth does not grow with the increasing number of IGWs if only non-overlapping channels are leveraged. The reason is that there are only three nonoverlapping channels in 2.4 GHz band, and at least two IGW-trees are assigned the same channel if more than three IGW-trees exist. With the POC-enabled channel assignment, the total utilized bandwidth improves obviously. Furthermore, in Scenario II (Fig. 17(b)), the total utilized bandwidth almost achieves linear growth with partiallyoverlapped channel assignment schemes. Hence, the total utilized bandwidth cannot be dramatically improved by simply increasing the number of the IGWs unless more spectrum resource is available, which implies that it is necessary to exploit partially-overlapped channels in virtual access network embedding. The total utilized bandwidth gains incurred by POC-enabled channel assignment is demonstrated in Fig. 18. When three IGWs are deployed in the WMN, the total utilized bandwidth gains incurred by partially-overlapped channels equal zero, since there are three non-overlapping channels in 2.4 GHz frequency band. When more than three IGWs exist in the WMN, the gains in both scenarios are remarkable.

Table 3 Default parameter settings. Physical meaning

Value

Population size Evolving generitions VAN number Bandwidth requirements of VANs Crossover probability (PC) Mutation probability (PM)

8 100 100 Uniformly distributed between 1 Mbps and 20 Mbps 0.5 0.9

Table 4 Detailed settings of Scenario II. MAP number

3 4 5 6 7 8

40 50 65 80 90 100

Total Utilized Bandwidth (Mbps)

IGW number

300 m  300 m 450 m  450 m 600 m  600 m 750 m  750 m 900 m  900 m 1000 m  1000 m

Total Utilized Bandwidth (Mbps)

Area size

300 250 200 150

EGA-POC GA-POC EGA-NOC GA-NOC

100 0

20

40

60

80

450 400 350 300 250 EGA-POC GA-POC EGA-NOC GA-NOC

200 150

100

0

20

40

Generations

60

80

100

Generations

(a) Scenario I

(b) Scenario II

500 450 400 350

Total Utilized Bandwidth (Mbps)

Total Utilized Bandwidth (Mbps)

Fig. 16. Convergence speed comparison between GA and EGA.

Capacity GR-NOC GR-POC GA-NOC GA-POC EGA-NOC EGA-POC

300 250 200 150 100

3

4

5

6

7

8

500 450 400 350

Capacity GR-NOC GR-POC GA-NOC GA-POC EGA-NOC EGA-POC

300 250 200 150 100

3

4

5

6

Number of IGWs

Number of IGWs

(a) Scenario I

(b) Scenario II

Fig. 17. Total utilized bandwidths of different algorithms in two scenarios.

7

8

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POC Gain

0.8

1

GR GA EGA

0.8

POC Gain

1

0.6 0.4 0.2

GR GA EGA

0.6 0.4 0.2

0

0 3

4

5

6

7

8

3

4

5

6

Number of IGWs

Number of IGWs

(a) Scenario I

(b) Scenario II

7

GR-NOC GR-POC

GA-NOC GA-POC

EGA-NOC EGA-POC

1 0.8 0.6 0.4 0.2

Resource Utilization Ratio

Resource Utilization Ratio

Fig. 18. POC gains in two scenarios.

0

GR-NOC GR-POC

0.8 0.6 0.4 0.2 0

GR-NOC GR-POC

GA-NOC GA-POC

(b) m = 4 EGA-NOC EGA-POC

1 0.8 0.6 0.4 0.2

Resource Utilization Ratio

Resource Utilization Ratio

EGA-NOC EGA-POC

1

(a) m = 3

0

GR-NOC GR-POC

GA-NOC GA-POC

EGA-NOC EGA-POC

1 0.8 0.6 0.4 0.2 0

GR-NOC GR-POC

GA-NOC GA-POC

1 0.8 0.6 0.4 0.2 0

(d) m = 6 EGA-NOC EvGA-POC

Resource Utilization Ratio

(c) m = 5 Resource Utilization Ratio

GA-NOC GA-POC

GR-NOC GR-POC

GA-NOC GA-POC

1 0.8 0.6 0.4 0.2 0

(e) m = 7

(f) m = 8

Fig. 19. Resource utilized ratios with different IGW numbers in Scenario I.

EGA-NOC EGA-POC

8

GR-NOC GR-POC

GA-NOC GA-POC

EGA-NOC EGA-POC

1 0.8 0.6 0.4 0.2 0

Resource Utilization Ratio

P. Lv et al. / Ad Hoc Networks 10 (2012) 1362–1378

Resource Utilization Ratio

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GR-NOC GR-POC

0.8 0.6 0.4 0.2 0

(b) m = 4 EGA-NOC EGA-POC

1 0.8 0.6 0.4 0.2 0

Resource Utilization Ratio

Resource Utilization Ratio

GA-NOC GA-POC

GR-NOC GR-POC

EGA-NOC EGA-POC

1

0.6 0.4 0.2 0

(d) m = 6 EGA-NOC EGA-POC

1 0.8 0.6 0.4 0.2 0

(e) m = 7

Resource Utilization Ratio

Resource Utilization Ratio

GA-NOC GA-POC

GA-NOC GA-POC

0.8

(c) m = 5 GR-NOC GR-POC

EGA-NOC EGA-POC

1

(a) m = 3 GR-NOC GR-POC

GA-NOC GA-POC

GR-NOC GR-POC

GA-NOC GA-POC

EGA-NOC EGA-POC

1 0.8 0.6 0.4 0.2 0

(f) m = 8

Fig. 20. Resource utilized ratios with different IGW numbers in Scenario II.

Detailed resource utilization ratios with various IGW numbers are illustrated in Fig. 19 (Scenario I) and Fig. 20 (Scenario II). Since solutions are generated randomly by GA, it cannot guarantee its resource utilization ratios to be higher than GR, e.g., the cases shown in Fig. 19(c), (f), Fig. 20(d), and (e). However, the performance of EGA can never be worse than GR because EGA uses GR solutions as initial chromosomes. The results indicate the effectiveness and the efficiency of EGA.

9. Conclusion To embed virtual access networks in WMNs to satisfy the end-to-end bandwidth demands of users, an OFDMAbased WMN architecture was designed. Under this architecture, a POC-enabled channel assignment scheme was proposed to utilize partial overlapping channels and support multiple virtual access networks simultaneously. Since the problem of virtual access network embedding is NP-hard, it was solved by an enhanced genetic algorithm with greedy strategies. Simulation results validated the effectiveness of the enhanced genetic algorithm and also demonstrated the feasibility of virtual network access in

WMNs. A distributed algorithm for virtual access network embedding is subject to future research. Acknowledgement The research work of Xudong Wang is supported by National Natural Science Foundation of China (NSFC) No. 61172066, Shanghai Pujiang Scholar Program 10PJ1406100, and the MOE Program for New Century Excellent Talents. The research work of Pin Lv and Ming Xu is supported by NSFC Nos. 61070211, 61070198, 60903040, and 61170288. The authors would like to thanks these sponsors for their generous support.

Appendix A. NP-completeness proof of channel assignment problem We use S to represent the channel assignment problem. S: Given a channel set C and a conflict graph G = (V, E) where jVj = n, find a channel vector hc1, c2, . . . , cni to satisfy that, for each eij 2 E, jci  cjj P w(eij), where ci 2 C (i = 1, . . . , n) and w(eij) is the least channel separation between node i and node j.

P. Lv et al. / Ad Hoc Networks 10 (2012) 1362–1378

Proof. (1) Given a channel vector ~ c, it can be determined in polynomial time whether ~ c is a solution of S. Thus, based on the definition of NP, it can be concluded that S 2 NP. (2) The notation S⁄ is used to denote the classical chromatic number problem. S⁄: Given a graph G and a positive integer k, find a function /: N ? Zk to satisfy that, if node u and v are adjacent, then /(u) – /(v). The problem S⁄ has been proved to be NP-complete in [29]. The problem S⁄ is a special case of S where the weight of each edge in S⁄ is set to 1. Hence, S 2 NP-hard. Since both S 2 NP and S 2 NP-hard hold, it can be proven that S 2 NP-complete. h References [1] I. Akyildiz, X. Wang, W. Wang, Wireless mesh networks: a survey, Computer Networks 47 (4) (2005) 445–487. [2] P. Pathak, R. Dutta, A survey of network design problems and joint design approaches in wireless mesh networks, IEEE Communications Surveys & Tutorials 13 (3) (2010) 396–428. [3] H. Zhu, M. Li, I. Chlamtac, B. Prabhakaran, A survey of quality of service in ieee 802.11 networks, IEEE Wireless Communications 11 (4) (2004) 6–14. [4] M. Yu, Y. Yi, J. Rexford, M. Chiang, Rethinking virtual network embedding: substrate support for path splitting and migration, ACM SIGCOMM Computer Communication Review 38 (2) (2008) 17–29. [5] Z. Cai, F. Liu, N. Xiao, Q. Liu, Z. Wang, Virtual network embedding for evolving networks, in: Proceedings of GLOBECOM, IEEE, 2010, pp. 1–5. [6] Y. Zhu, M. Ammar, Algorithms for assigning substrate network resources to virtual network components, Proceedings of IEEE INFOCOM, vol. 2, IEEE, 2006, pp. 1–12. [7] N. Chowdhury, M. Rahman, R. Boutaba, Virtual network embedding with coordinated node and link mapping, in: Proceedings of INFOCOM, IEEE, 2009, pp. 783–791. [8] J. Lischka, H. Karl, A virtual network mapping algorithm based on subgraph isomorphism detection, in: Proceedings of the 1st ACM Workshop on Virtualized Infrastructure Systems and Architectures, ACM, 2009, pp. 81–88. [9] T. Anderson, L. Peterson, S. Shenker, J. Turner, Overcoming the internet impasse through virtualization, Computer 38 (4) (2005) 34–41. [10] N. Chowdhury, R. Boutaba, Network virtualization: state of the art and research challenges, IEEE Communications Magazine 47 (7) (2009) 20–26. [11] N. Chowdhury, R. Boutaba, A survey of network virtualization, Computer Networks 54 (5) (2010) 862–876. [12] J. Lu, J. Turner, Efficient mapping of virtual networks onto a shared substrate, Tech. Rep., Washington University, USA, 2006. [13] I. Houidi, W. Louati, D. Zeghlache, A distributed virtual network mapping algorithm, in: IEEE International Conference on Communications, 2008 (ICC’08), IEEE, 2008, pp. 5634–5640. [14] M. Rahman, I. Aib, R. Boutaba, Survivable virtual network embedding, Networking 2010 (2010) 40–52. [15] S. Paul, S. Seshan, Virtualization and slicing of wireless networks, in: Technical Report GENI Design Document 06-17, GENI Wireless Working Group, 2006, pp. 1–17. [16] K. Park, C. Kim, A framework for virtual network embedding in wireless networks, in: Proceedings of CFI, ACM, 2009, pp. 5–7. [17] R. Matos, C. Marques, S. Sargento, K. Hummel, H. Meyer, Analytical modeling of context-based multi-virtual wireless mesh networks, Ad Hoc Networks (2011) 1–19. [18] R. Matos, S. Sargento, K. Hummel, A. Hess, K. Tutschku, H. de Meer, Context-based wireless mesh networks: a case for network virtualization, Telecommunication Systems (2011) 1–14. [19] S. Shrestha, J. Lee, S. Chong, Virtualization and slicing of wireless mesh network, in: Proceedings of CFI, ACM, 2009. [20] G. Smith, A. Chaturvedi, A. Mishra, S. Banerjee, Wireless virtualization on commodity 802.11 hardware, in: Proceedings of

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Pin Lv is currently a student pursuing his Ph.D. His research interests include wireless mesh networks, network virtualization, and cloud computing.

Xudong Wang is currently with UM-SJTU Joint Institute, Shanghai Jiao Tong University, Shanghai, China. He is also an affiliate faculty member with the Electrical Engineering Department at the University of Washington. Since he received the Ph.D. degree in Electrical and Computer Engineering from Georgia Institute of Technology in August 2003, he has been working as a senior research engineer, senior network architect, and R&D manager in several companies. He has been actively involved in R&D, technology transfer, and commercialization of various wireless networking technologies. His research interests include low-power radio architecture and protocol suite, deep-space network architecture and protocols, cognitive/software radios, LTE-A, wireless mesh networks, cross-layer design, wireless sensor networks, and ultra-wideband networks. He holds several patents on wireless networking technologies and most of his inventions have been successfully transferred to products. He is an editor for Elsevier Ad Hoc Networks and ACM/Kluwer Wireless Networks. He was also a guest editor for several journals. He was the demo co-chair of the ACM International Symposium on Mobile Ad Hoc Networking and Computing (ACM MOBIHOC 2006), a technical program co-chair of Wireless Internet Conference (WICON) 2007, and a general co-chair of WICON 2008. He has been a technical committee member of many international conferences and a technical reviewer for numerous international journals and conferences. He is a senior member of IEEE and was a voting member of IEEE 802.11 and 802.15 Standard Committees. He received his B.E. degree in Electric Engineering and his first Ph.D. degree in Automatic Control in 1992 and 1997, respectively, from Shanghai Jiao Tong University, Shanghai, China.

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Ming Xu is currently a professor at the National University of Defense Technology. He is the head of the Department of Network Engneering. He is a senior member of China Computer Federation (CCF), member of ACM, IEEE. His recent research interests include wireless mesh networks, mobile security, wireless sensor networks, and mobile data management. He has published over 130 academic papers in journals and conferences, co-authored of 3 books. Among them, Mobile Computing Technology is the first book in China mainland containing extensive research results around wireless networks and mobile computing. He is an editor of IASTED International

Journal of Computers & Applications, also editor of Journal of Communication. He has co-chaired 4 international conferences, and been program committee members for over 30.