Joint bandwidth allocation, element assignment and scheduling for wireless mesh networks with MIMO links

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Computer Communications 31 (2008) 1372–1384 www.elsevier.com/locate/comcom

Joint bandwidth allocation, element assignment and scheduling for wireless mesh networks with MIMO links Jun Wang a,b,c,*, Peng Du a,b,c, Weijia Jia b,c, Liusheng Huang a,c, Huan Li d a

Department of Computer Science, University of Science and Technology of China, 96#, Jinzhai Road, Hefei, Anhui 230026, China b Department of Computer Science, City University of Hong Kong, Hong Kong, China c Joint Research Lab, CityU-USTC Advanced Research Institute, Suzhou, China d Department of Computer Science, Beihang University, Beijing, China Available online 5 February 2008

Abstract With the unique features of spatial multiplexing and interference suppression, Multiple Input Multiple Output (MIMO) techniques have great potential in the improvement of network capacity over conventional antenna technologies. In order to exploit the benefit of simultaneous transmissions provided by MIMO, researchers have proposed a number of cross-layer optimizations and MAC layer designs to increase the throughput of wireless mesh or ad hoc networks, where the number of elements in the antenna arrays are preallocated or evenly assigned to the routers. In this paper, we argue that using the same number of elements in each antenna array in all routers is not a necessary condition for the improvement of system performance. This is because the requirement for the number of elements is quite different for each router. Especially at those critical routers that have huge aggregate traffic toward the gateway, more elements are needed not only for the traffic relay but also for the interference suppression. Based on this observation, we define the joint problem of bandwidth allocation, element assignment and scheduling to characterize the throughput benefits of cross-layer optimizations. We propose a Cost-Aware Element Assignment (CAEA) technique to minimize the total number of the antenna elements when still achieving the optimal bandwidth allocation. In addition, to verify the efficiency of the CAEA assignment, a heuristic Traffic-aware Stream-controlled Link Scheduling (TSLS) algorithm is proposed to provide a schedulable bandwidth allocation. We demonstrate through extensive simulations that our solutions (CAEA, TSLS) not only effectively save the total cost on antenna elements but also perform close to optimal on the average. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Wireless mesh networks; MIMO; Bandwidth allocation; Element assignment; Scheduling

1. Introduction Recent years, Wireless Mesh Networks (WMNs) [1] have emerged as a promising technology that provides wireless broadband accessibility and thus, extends the Internet connectivity at the edge and improves the network coverage and economy efficiency. A typical deployment of

* Corresponding author. Address: Department of Computer Science, University of Science and Technology of China, 96#, Jinzhai Road, Hefei, Anhui 230026, China. Tel.: +86 512 87161297; fax: +86 512 87161381. E-mail addresses: [email protected] (J. Wang), [email protected] (P. Du), [email protected] (W. Jia), [email protected] (L. Huang), [email protected] (H. Li).

0140-3664/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2008.01.053

a multi-hop infrastructure-based WMN consists of three major components: base station (or so called gateway), a set of mesh routers and mesh clients. The base station is presented that is wired to a larger network (e.g., the Internet); a set of mesh routers integrate traffic flows for local (one hop) clients and relay communication through wireless channel in order to transfer data to or from the Internet through the gateways; a large number of mesh clients can be any of the mobile devices such as laptops, PDAs and smartphones. Mesh routers are normally stationary and therefore, have less topology change, node failure and energy constraints. However, the use of WMN as a backbone for large wireless access networks imposes high bandwidth requirements. At the same time, the interference

J. Wang et al. / Computer Communications 31 (2008) 1372–1384

among simultaneous transmissions may dramatically cause capacity reduction [2]. The ability of using Multiple Input Multiple Output (MIMO) antenna technology in the wireless transmission is one of the major techniques to alleviate the problem, and thus improve throughput substantially [3,4]. In a typical MIMO transmission system as shown in Fig. 1, the transmitter and the receiver are equipped with multiple antenna elements. The data are first encoded and mapped to complex modulation symbols (such as QPSK, M-QAM) which produce several separate symbol streams. Each stream is then mapped onto one of the multiple transmitting antenna elements. After upward frequency conversion, filtering and amplification, the signals are launched into the wireless channel. At the receiver, the signals are captured by multiple receiving antenna elements and demodulation and demapping operations are performed to recover the data. The presence of multiple elements at both ends of a link creates multiple independent simultaneous transmissions, and thus achieve higher throughput over conventional communication systems. The MIMO links have the following characteristics: 1. Spatial multiplexing: Multiple independent data streams can be transmitted simultaneously to one or more receivers to provide extremely high spectral efficiencies (increase in capacity) at the cost of no extra bandwidth or power [5]. 2. Interference suppression: MIMO link can suppress interference from neighboring links as long as the total number of useful streams and interfering streams are no greater than that of receiving antenna elements [6]. 3. Half-duplex: The antenna array cannot be in the mode of transmission and reception at the same time, although it can communicate in multiple streams with more than one node. 4. Diversity: Dependent streams can be transmitted on multiple elements to achieve transmitting diversity gain [7]. This diversity gain can provide us with range extension (a larger transmission range) or power minimization, or better link reliability as desired. Due to above properties, the MIMO technology has a great potential to improve the throughput in traffic-intensive wireless networks. Recent years, advances in MIMO links have become extremely popular and made their way into the WLAN (IEEE 802.11n), WIMAX (802.16) and

Space-time coding data

1

1

2

2

Demapping Demodulation

Modulation Mapping

M

N

Decoding

Fig. 1. MIMO wireless transmission system.

data

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3G networks. Several vendors such as Sohoware are also well motivated to offer MIMO equipments for commercial use in daily life. In order to fully exploit this new technology, researchers have done great work on the throughput optimization problems from routing, scheduling and MAC protocol design aspects. However, most of them pre-assume each router has been equipped with equal number of elements that is called k-element architecture. In this paper, we argue that evenly assigning the elements to every node may not be necessary or effective to satisfy the whole network traffic on demand. This is because not only the peer-to-peer wireless interference, but also the element assignment on each node will influence the WMN capacity [2,3]. If we want to achieve the performance in both throughput and fairness, meanwhile minimize the total number of elements, the problem becomes much harder and thus novel techniques are required. In this paper, we study the problems and relationships between bandwidth allocation, element assignment and scheduling with the objective of optimizing the throughput and fairness and meanwhile minimizing the total cost on the antenna elements in WMNs with MIMO links. Here, element assignment means specifying a certain number of elements for each node. The major contributions of this paper are as follows.  We analyze and formulate all kinds of system constraints on the traffic flows and element assignment for WMNs/MIMO. We well define the bandwidth allocation, element assignment and scheduling problems as well as the in-deep relationships among them.  We show that the problems of bandwidth allocation, element assignment and scheduling can be analyzed and processed in an interactive way. In addition to the optimal solutions (by formulating the linear programming problem), we propose novel efficient heuristics to address those problems. We demonstrate through extensive experiments the proposed heuristics can achieve a performance close to optimal solutions.  To the best of our knowledge, this paper for the first time investigates the element assignment problem together with the bandwidth allocation problem for WMNs with MIMO links. The rest of the paper is organized as follows. We state the system model in Section 2. We present detailed problem definitions in Section 3. We describe our optimal solutions for bandwidth allocation in Section 4 and element assignment in Section 5. To verify and further guarantee the feasibility of our solutions, we provide a heuristic scheduling algorithm in Section 6. The implementation issues of the proposed solutions are discussed in Section 7. Section 8 shows the simulation results. We present related work in Section 9. Finally, we draw our conclusions and discuss future work in Section 10.

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2. System model In this section, we present background information and assumptions on MIMO-based network architecture and the system model we will use for the rest of the paper. 2.1. Network architecture The system is considered as a multi-hop Infrastructure Wireless Mesh Network (IWMN) that consists of several fixed wireless mesh routers. The routers provide network connectivity to end-user mobile clients within their coverage area. The wireless mesh routers themselves form a multi-hop wireless backbone for relaying the traffic to and from the clients. One router together with all its clients is considered as one node in this paper. Some of the wireless mesh routers are designated as gateways that provide functionality to enable direct connection to a large network such as the Internet. We model the backbone of this IWMN as a directed graph G ¼ ðV ; EÞ; where V represents the set of nodes in the network and E represents the set of directed links (transmissions). The set of gateway nodes is represented as V 1 and we denote the non-gateway nodes by the set V 2 ðV ¼ V 1 [ V 2 Þ. We suppose all the traffic within the mesh is either from the user devices to the outer network or vice versa. Here, we abstract the outer network as a super node (sink or source) which connects all the gateways of the WMN. Because the traffic is relayed to or from the super node, the paths taken by the traffic flows are likely to form a tree structure in which the super node is the ‘‘root” and the user devices are the ‘‘leaves”. Here, we can simply use the Breadth-First Search (BFS) to construct a Minimum Spanning Tree (MST) T for the new graph G0 with the super node. Then we remove the super node from the resulting MST T. At the end of this process, we will obtain several non-intersect routing trees, in which different gateways are different roots. For each node except the gateways, the traffic can be separated as uplink and downlink flows. For ease of exposition, in this paper, we will address the problems for all uplink flows, and this can be extended to bidirectional link easily if we consider the uplink and downlink separately. And this technique has been applied to the MAC layer design of 802.16 protocols. In the rest of the paper, if a transmission is from node u to node v, then v is called the parent node of u, denoted as PARðuÞ; and the children set of node u is represented as CHIðuÞ. Also, we denote T ðuÞ as the sub tree of T with root of node u.

other if they stand within the interference range of one another. In this paper, we assume all transmission ranges (denoted as RT ) are same and the interference range is greater than transmission range. Suppose all interference ranges are same and denoted as RI , we say node v is node u’s ‘‘neighbor” if node v is in the interference range of node u. The set of neighbors of node u is defined as NEIðuÞ. 2.3. Abstraction of MIMO links We use the following abstraction and assumptions for the physical layer in our work. Each node u (u 2 V ) being capable of forming MIMO links with its neighbors may have multiple antenna elements, denoted as IðuÞ (1 6 IðuÞ 6 k). Here, k is the maximum possible number of elements for each node. A node with IðuÞ antenna elements is also referred to as a node with IðuÞ Degree of Freedoms (DOFs). In a MIMO link, a transmitter u can use up all its available DOFs to send at most IðuÞ independent streams to one or more receivers simultaneously. This constraint is called Transmission DOF Constraint. At the receiver end, a receiver v can isolate and decode all the incoming streams successfully as long as the total number of receiving streams (n) including the interference is less than or equal to its DOFs (n 6 IðvÞ). On the other hand, if the total receiving streams overwhelm the DOFs at the receiver (n > IðvÞ), it will not be possible to decode any of the desired signal streams. This is called Reception DOF Constraint. Here, we do not consider partial interference suppression [8] whereby fewer degree of freedoms are needed if the interference is low. This is because in the 802.11 settings, partial suppression does not perform well [9]. Since not all streams have the same capacity [10], we put the streams from the same node in a non-increasing order, for instance, stream1 P stream2 P    P streamk. We denote by Cðu; iÞ the capacity for link u ! PARðuÞ, transmitting on stream i. Take the network in Fig. 2 for example, we assume the capacity set C ¼ fCðu; 1Þ ¼ 1; Cðu; 2Þ ¼ 0:8; Cðu; 3Þ ¼ 0:6ju 2 fa; c; egg and the element assignment is fIðuÞ ¼ 3; u 2 fa; b; c;d; e; f gg. Since the links interfere with each other, the total number of streams transmitted simultaneously cannot be greater than three. If the links are scheduled in a TDMA fashion (i.e., nodes a, c and e transmit in turn), then the total throughput is

Non-gateway node Independent stream

1

2

3

Interferenece Antenna element

e d

1

2.2. Wireless transmission and interference model

2

a

To accomplish a transmission, the receiver has to be within the transmission range of the sender, using a common channel assigned to their antenna elements. Two pairs of nodes using the same channel may interfere with each

f

Gateway node

3

1

b

2 3

c

Fig. 2. A topology example with three antenna elements in each node.

J. Wang et al. / Computer Communications 31 (2008) 1372–1384

2.4. However, if the best stream is selected for transmission from each link (MIMO), then the total throughput is 3. Here, how to select streams for transmission is also called Stream Control [11]. 3. Constraints and problem definitions In this section, we first study the details of transmission from the aspect of link scheduling. Then we abstract the constraints on the traffic flows that characterize MIMO links. Based on these constraints, we formalize the problem definitions with the goal of cross-layer optimization on the bandwidth allocation and element assignment.

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P S that is 16s6S X u;i;s 6 2. After introducing this constraint to traffic flow according to Eq. (1), we obtain the Stream Capacity Constraint: 1 f ðu; iÞ 6 Cðu; iÞ; 2

8u 2 V 2 ;

16i6k

Now let us consider the DOF constraints. First, we define the set of interfering nodes as INT ðuÞ, when node u is receiving packets. Any node v in INT ðuÞ must satisfy two conditions: (1) node v is in the interference range of node u, that is, v 2 NEIðuÞ; (2) when node u is in reception, node v is in the transmission mode: ðhopðvÞ  hopðuÞÞmod2 ¼ 1. So we have: INT ðuÞ ¼ fvjv 2 NEIðuÞ; ðhopðvÞ  hopðuÞÞmod2 ¼ 1g

3.1. Constraints In order to address the relationship between traffic flow and link scheduling, we assume all nodes are synchronized and will transmit packets in time slots according to a centralized scheduling. We introduce two indicators X u;i;s and Y u;i;s . Let X u;i;s be 1 if node u transmits in time slot s on stream i, otherwise 0; in the same way, let Y u;i;s be 1 if node u receives packets in time slot s on stream i, otherwise 0. We use f ðuÞ to denote the outgoing traffic rate for node u, and f ðu; iÞ the traffic rate on stream i, so for node u, the f ðu;iÞ . We can also compute utilization ratio for stream i is Cðu;iÞ the utilization ratio from a microscopic view. In a schedule of S timePslots, the stream i can be used for transmission in total of 16s6S X u;i;s time slots, then the utilization ratio is P X u;i;s 16s6S . Hence, the relationship between link scheduling S and traffic flow is represented as: P f ðu; iÞ 16s6S X u;i;s ð1Þ ¼ Cðu; iÞ S Since MIMO link is half-duplex, a node cannot transmit and receive packets at the same time. For the ease of explanation, we assume that each node receives and transmits alternatively. That is, if node u transmits packets at time slot s, then it should be in the mode of reception at time slot s þ 1; meanwhile, nodes in set PARðuÞ [ CHIðuÞ receive at time slot s and then are allowed to transmit at time slot s þ 1. This assumption is reasonable since most nodes in WMN are routers to relay the traffic and the transmission and reception time are almost equal in the long term. To this end, at one time slot, a node is receiving or transmitting can be pre-determined by the number of hops from its gateway. Here, we define the number of hops from node u to its gateway node as hopðuÞ. As we mentioned before, for each node u, different streams have different capacities and we assume the capacity for stream i ð1 6 i 6 kÞ is Cðu; iÞ with a non-increasing order. That is Cðu; iÞ P Cðu; i þ 1Þ, and we assume Cðu; iÞ ¼ 0 if IðuÞ < i 6 k. Due to the half-duplex constraint, for a node u, at most half of the total time slots can be used for transmission,

ð2Þ

ð3Þ

When arranged to transmit packets at time slot s, a node u can P send at most IðuÞ independent streams simultaneously ð 16i6k X u;i;s 6 IðuÞ); and at the same time it cannot receive P P P any packets: 16i6k Y u;i;s ¼ v2CHIðuÞ 16i6k X v;i;s ¼ 0. According to our definition of INT ðuÞ, the nodes in INT ðuÞ are in reception when node u is transmitting. So there is P no stream sent from these nodes. We have: P v2INT ðuÞ 16i6k X v;i;s ¼ 0. If we denote the set of time slots as ST (jST j ¼ 12 S) in which node u is allowed to transmit, we have: 8 ! > P P P > > < X v;i;s ¼ 0 s2ST v2CHIðuÞ[INT ðuÞ 16i6k ð4Þ P P > > > X u;i;s 6 12 S  IðuÞ : s2ST 16i6k

Similarly, when a node u is receiving, the total number of incoming streams from its children and interference nodes P should not be greater than IðuÞ ð v2CHIðuÞ[INT ðuÞ P 16i6k X v;i;s 6 IðuÞÞ, and node u cannot transmit at the same time (X u;i;s ¼ 0Þ. If we denote SR (jSRj ¼ 12 S) as the set of time slots in which node u is in the mode of reception, we have: 8 ! > P P P > > < X v;i;s 6 12 S  IðuÞ s2SR v2CHIðuÞ[INT ðuÞ 16i6k ð5Þ P P > > > X u;i;s ¼ 0 : s2SR 16i6k

From Eqs. (4) and (5), in a schedule of S time slots, the total number of streams from node u should not be greater than 12 S  IðuÞ, that is: X X X X X X X u;i;s ¼ X u;i;s þ X u;i;s 16s6S 16i6k

s2ST 16i6k

1 6 S  IðuÞ 2

s2SR 16i6k

ð6Þ

Now, according to Eq. (1), we can deduce the Transmission DOF Constraint on the traffic flow: X f ðu; iÞ 1 6 IðuÞ; 8u 2 V 2 ð7Þ Cðu; iÞ 2 16i6k

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Also from Eqs. (4) and (5), in total S time slots, the number of streams from nodes in set CHIðuÞ [ INT ðuÞ should not be greater than 12 S  IðuÞ: ! X X X 1 ð8Þ X v;i;s 6 S  IðuÞ 2 16s6s v2CHIðuÞ[INT ðuÞ 16i6k Change Eq. (8) to constraint on traffic flow, we have the Reception DOF Constraint: X v2CHIðuÞ[INT ðuÞ

X f ðv; iÞ 1 6 IðuÞ; Cðv; iÞ 2 16i6k

GuarBW ðFBAÞ ¼ minfgðuÞ; u 2 V 2 g 8u 2 V

16i6k

P Here, the system throughput is: u2V 2 gðuÞ. Table 1 summarizes all constraints on the traffic flow we discussed before. In the rest of the paper, we will use the index number of those constraints instead of naming the detailed constraints. 3.2. Problem definitions In this section, we will consider three closely correlated problems: bandwidth allocation, element assignment and scheduling: 1. Bandwidth allocation: If given an element assignment fIðuÞ; u 2 V g, how to allocate the bandwidth can maximize the total throughput meanwhile achieving fairness for each node? 2. Element assignment: If given a bandwidth allocation for each node, what is minimum requirement for the total number of antenna elements, in another word, how to specify the number of elements in each node can save the cost most and satisfy the bandwidth allocation? 3. Scheduling: Given a bandwidth allocation and an element assignment for each node, how to obtain a feasible schedule for the allocated bandwidth? Based on the abstraction on the constraints (1–4), these three kinds of problems can be formally defined as follows. Table 1 Constraints definitions Index Constraint Constraint Constraint Constraint

(1) (2) (3) (4)

ð11Þ

ð9Þ

For each non-gateway node u, the total incoming traffic plus the traffic from the clients of this node must equal to the total outgoing traffic. We denote gðuÞ as the total allocated bandwidth to the clients of node u. Then we have Flow Balance Constraint: X X X f ðv; iÞ þ gðuÞ ¼ f ðu; iÞ; 8u 2 V 2 ð10Þ 16i6k v2CHIðuÞ

Problem Definition 1. Feasible Bandwidth Allocation Problem (FBAP): For each node u, given an EA (Element Assignment) fIðuÞ; u 2 V g, if the allocated bandwidth fgðuÞ; u 2 V 2 g can satisfy Constraints (1), (2), (3) and (4), then we can achieve feasible traffic flows for each node; the bandwidth allocation problem is called FBAP; the bandwidth allocation we obtained fgðuÞ; u 2 V 2 g is called a FBA. We define the guaranteed bandwidth of this FBA as GuarBW(FBA), which is:

Description

Formulation

Stream Capacity Constraint Transmission DOF Constraint Reception DOF Constraint Flow Balance Constraint

Eq. Eq. Eq. Eq.

(2) (7) (9) (10)

Problem Definition 2. Feasible Element Assignment Problem (FEAP): For each non-gateway node u, given a FBA fgðuÞ; u 2 V 2 g, we should decide the element number IðuÞ for each node so that we can achieve a feasible traffic flows that satisfy Constraints (1), (2), (3) and (4). This problem is called FEAP. The element assignment we obtained fIðuÞ; u 2 V g is called a FEA. If we define the cost of this FEA as costðFEAÞ, then we have: costðFEAÞ ¼

X

IðuÞ

ð12Þ

u2V

Problem Definition 3. Schedulable Bandwidth Allocation Problem (SBAP): For each node u, if given the element number IðuÞ and the allocated bandwidth gðuÞ, the problem of how to schedule the bandwidth requests in each time slots is called SBAP. Assume there are M time slots per second. If bandwidth allocation fgðuÞ; u 2 V 2 g can be scheduled in a period of S time slots, then we can obtain a 0 Schedulable Bandwidth Allocation (SBA) fgðuÞ ¼ kgðuÞ; u 2 V 2 g, in which k ¼ MS : Problem Definition 4. Maximum Guaranteed Feasible Bandwidth Allocation Problem (MGFBAP): A bandwidth allocation fgðuÞ; u 2 V 2 g is a MGFBA if and only if it satisfies two conditions: (1) fgðuÞ; u 2 V 2 g is a FBA; (2) minfgðuÞ; u 2 V 2 g ¼ maxfGuarBW ðFBAÞ; 8FBAg:

Table 2 Notations Notations

Meanings

V V1 V2 PAR(u) CHI(u) NEI(u) T(u) INT(u)

Set of nodes in the WMN Set of gateway nodes in the WMN Set of non-gateway nodes in the WMN Parent of node u Children of node u Neighbors of node u Sub tree with the root of node u Set of nodes whose transmission will interfere node u’s reception Bandwidth request from node u’s end clients Bandwidth allocation to node u’s end clients Total traffic out from node u to its parent Traffic out from node u to its parent on stream i Number of elements assigned for node u Maximum number of elements for node u Capacity on stream i from node u

req(u) g(u) f(u) f(u,i) I(u) k c(u,i)

J. Wang et al. / Computer Communications 31 (2008) 1372–1384

Problem Definition 5. Minimum Cost Feasible Element Assignment Problem (MCFEAP): An element assignment fIðuÞ; u 2 V g is a MCFEA if and only if it satisfies (1) fIðuÞ; P u 2 V g is a FEA; (2) u2V IðuÞ ¼ minfcostðFEAÞ; 8FEAg:

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capacity; the solution of LP1 (a) has a linear relationship with k. That is: Suppose IðuÞ ¼ k; 8u 2 V ; Cðu; iÞ ¼ c; 8 u 2 V 2 ; 1 6 i 6 k. We have a ¼ k  q, where q is irrelevant to k.

The notations used in this paper are listed in Table 2. 4. Bandwidth allocation In this section, we will solve the problem of bandwidth allocation in WMNs with MIMO links. We consider two fairness modes: max–min fairness and proportional fairness. The former tries to provide maximum bandwidth guarantees for all users in the WMNs, which is very important for those real-time multimedia applications [12]; while the latter achieves proportional bandwidth allocation with respect to the user’s requests. The benefit for proportional fairness model can be found in [13], for which we will not discuss any more. In the following subsections, we first formulate the bandwidth allocation problem into two Linear Programming (LP) problems [14] under the two fairness modes, respectively. Then we can find the optimal solution by solving the LP formulas. How to solve the LP problems have been thoroughly studied in the literature. The computational complexity of such problems is shown to be polynomial [15]. In addition, we provide a heuristic solution for the bandwidth allocation in Section 6 with a time complexity of Oðn4 Þ (n is number of nodes in WMN). 4.1. Max–min fair allocation In this fairness mode, we try to find a MGFBA as defined in Section 3 where Constraints (1), (2), (3) and (4) should be satisfied with a maximum guaranteed bandwidth for each node. The related LP formulation is depicted in LP1. Given an element assignment fIðuÞj1 6 IðuÞ 6 k; u 2 V g, LP1 provides a tight upper bound of the guaranteed bandwidth. LP 1 for MGFBA problem : Max a

v2T ðuÞ

Note that T ðuÞ is the sub tree whose root is node u. Since gðuÞ P a; 8u 2 V 2 We have: f ðuÞ ¼

P

v2T ðuÞ gðvÞ

P a  T ðuÞ

From Constraints (2), (3) and f ðuÞ ¼ have: 81 < 2 c  k P a  T ðuÞ : 12 c

k Pa

P

16i6k f ðu; iÞ,

we

8u 2 V 2 P

T ðvÞ

8u 2 V

ð14Þ

v2CHIðuÞ[INT ðuÞ

Solving Eq. (14), we can get:   1 1 ju 2 V 2 ; a 6  c  k  min min 2 T ðuÞ ( )! 1 ju 2 V min P v2CHIðuÞ[INT ðuÞ T ðvÞ

ð15Þ

If we define fg as: ( )!  1 1 fg ¼ min min ju 2 V ju 2 V 2 ; min P T ðuÞ v2CHIðuÞ[INT ðuÞ T ðvÞ 

ð16Þ

Subject to : gðuÞ P a; 8u 2 V 2 1 f ðu; iÞ 6 Cðu; iÞ; 8u 2 V 2 ; 1 6 i 6 k 2 X f ðu; iÞ 1 6 IðuÞ; 8u 2 V 2 Cðu; iÞ 2 16i6k X X f ðv; iÞ 1 6 IðuÞ; 8u 2 V Cðv; iÞ 2 v2CHIðuÞ[INT ðuÞ 16i6k X X X f ðv; iÞ þ gðuÞ ¼ f ðu; iÞ; 8u 2 V 2 v2CHIðuÞ 16i6k

Proof. For each non-gateway node u, we can easily deduce the traffic flow in node u according to the aggregate traffic model. X f ðuÞ ¼ gðvÞ ð13Þ

16i6k

Proposition 1. In the WMN with all nodes equipped with k antenna elements, if we assume all streams have the same

Note that fg is decided by the network topology and is not relevant to k.   Then we have a ¼ 12  c  fg  k, so the maximum guaranteed bandwidth has a linear relationship with k.

4.2. Proportional fair allocation In this part, we consider proportional fairness model and allocate the bandwidth according to the bandwidth request from the end clients proportionally. Here, for a non-gateway node u, we suppose the aggregate bandwidth request from node u’s end users is reqðuÞ. We formulate LP2 to provide a max–min proportional coefficient k.

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LP 2 for maximum proportional fair bandwidth allocation : Max k Subject to : gðuÞ P k  reqðuÞ; 8u 2 V 2 1 f ðu; iÞ 6 Cðu; iÞ; 8u 2 V 2 ; 1 6 i 6 k 2 X f ðu; iÞ 1 6 IðuÞ; 8u 2 V 2 Cðu; iÞ 2 16i6k X X f ðv; iÞ 1 6 IðuÞ; 8u 2 V Cðv; iÞ 2 v2CHIðuÞ[INT ðuÞ 16i6k X X X f ðv; iÞ þ gðuÞ ¼ f ðu; iÞ; 8u 2 V 2 v2CHIðuÞ 16i6k

16i6k

When considering transmission, we have Constraint (1): f ðu; iÞ 6 12 Cðu; iÞ. Remember that Cðu; iÞ is in a nonincreasing order, so when we use stream-controlled transmission, the streams with larger capacity will be used first. If we assume that the traffic can be fully allocated in the first gu streams, then our goal is to minimize gu : (  ) X 1  Cðu; iÞ  f ðuÞ P 0 ; 8u 2 V 2 ð18Þ gu ¼ min g 16i6g 2 Since gateway nodes can directly connect to the wired network, we have gu ¼ 0;

8u 2 V 1

Then the transmission requirement for node u is: IðuÞ P gu ;

Proposition 2. In the WMN with all nodes equipped with k antenna elements, if we assume all the streams have the same capacity; then the result k of LP2 has a linear relationship with k. That is: Suppose IðuÞ ¼ k; 8u 2 V ; Cðu; iÞ ¼ c; 8u 2 V 2 ; 1 6 i 6 k. We have k ¼ k  p, where p is irrelevant with k. Proof. (We use the same method as in the proof of Proposition 1, and here we omit the details).

5. Antenna element assignment In this section, we will focus on the element assignment problem. According to the analysis on the traffic and interference model, we propose a Cost-Aware Element Assignment (CAEA) algorithm to provide an optimal MCFEA which gives a tight lower bound of the total number of elements in the WMN. 5.1. An optimal solution of element assignment problem When the topology of the WMN has been settled, first we suppose all nodes are evenly assigned with k elements. Since k is the maximum possible number of elements equipped for each node, this k-element assignment provide the upper bound of all FBA under the conditions of different FEAs. Then we can calculate this optimal FBA by solving LP1 or LP2 according to different fairness model. We use this FBA as an input to find the MCFEA. 5.1.1. The DOF requirement for transmission First, we consider the DOF requirement for transmission in which we try to use as fewer streams as possible to fully allocate the outgoing traffic. Given a FBA as fgðuÞ; u 2 V 2 g, we can calculate the traffic for each nongateway node u in the following way: X X f ðuÞ ¼ gðvÞ ¼ f ðu; iÞ ð17Þ v2T ðuÞ

16i6k

ð19Þ

8u 2 V

ð20Þ

5.1.2. The DOF requirement for reception Before analyzing the DOF requirement for reception, we first define two new notations xu and hu , where xu represents the DOF requirement to suppress the interference from node u and can be defined as: X f ðu; iÞ xu ¼ ð21Þ ; 8u 2 V 2 1 Cðu; iÞ 16i6g 2 u

Then we define hu as the utilization ratio of the stream gu of node u: P f ðuÞ  16i6gu 1 12 Cðu; iÞ f ðu; gu Þ hu ¼ 1 ¼ ; 8u 2 V 2 1 Cðv; gu Þ Cðv; gu Þ 2 2 ð22Þ From the definitions of gu , xu and hu , we can easily get the relationships between them:  gu ¼ dxu e ð23Þ hu ¼ xu þ 1  gu When consider reception, we have Constraint (3): X f ðv; iÞ X IðuÞ P ; 8u 2 V 1 Cðv; iÞ v2CHIðuÞ[INT ðuÞ 16i6k 2 Since

f ðu;iÞ 1 Cðu;iÞ 2

IðuÞ P

ð24Þ

¼ 0; 8gu < i 6 k; u 2 V 2 , then we can get: X X xv ¼ ðgv  1 þ hv Þ ð25Þ

v2CHIðuÞ[INT ðuÞ

v2CHIðuÞ[INT ðuÞ

So the reception requirement for IðuÞ is: & ’ X IðuÞ P ðgv  1 þ hv Þ

ð26Þ

v2CHIðuÞ[INT ðuÞ

5.1.3. The DOF requirement for both transmission and reception The feasible element assignment should satisfy both transmission (Eq. (20)) and reception (Eq. (26)) requirements, then we have the MCFEA:

J. Wang et al. / Computer Communications 31 (2008) 1372–1384

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Table 3 Comparison between two FEAs for the example topology in Fig. 2

Stream capacity Bandwidth allocation GuarBW Element assignment Cost

FEA1

FEA2

54 Mbps gðuÞ ¼ 27 Mbps; u 2 fa; c; eg 27 Mbps IðuÞ ¼ 3; u 2 fa; b; c; d; e; f g 18 elements

54 Mbps gðuÞ ¼ 27 Mbps; u 2 fa; c; eg 27 Mbps IðuÞ ¼ 1; u 2 fa; c; e; f g; IðbÞ ¼ 3; IðdÞ ¼ 2 9 elements

Table 4 Algorithm 1. Cost-Aware Element Assignment (CAEA)

5.2. Algorithm of Cost-Aware Element Assignment

Input: Network topology graph GðV ; EÞ, Gateway nodes V 1 , Non-gateway nodes V 2 , Bandwidth allocation fgðuÞ; 8u 2 V 2 g, Stream capacity fCðu; iÞ; 8u 2 V 2 ; 1 6 i 6 kg Output: Element assignment fIðuÞ; 8u 2 V g Step 1: Compute traffic f ðuÞ; 8u 2 V 1: 8u 2 V 1 , f ðuÞ 0 P 2: 8u 2 V 2 , f ðuÞ v2T ðuÞ gðvÞ Step 2: Compute gu and hu , 8u 2 V 3: for each u 2 V 4: gu 0; 5: traffic f ðuÞ 6: do while traffic > 0 7: traffic traffic  12 Cðu; gu þ 1Þ 8: gu gu þ 1 traffic 1 þ 1Cðu;g 9: hu Þ

We present the CAEA algorithm based on the analysis of Section 5.1. We solve the problem into 3 steps (as shown in Table 4): 1. Calculate the traffic f ðuÞ for each node according to Eq. (17); 2. Compute gu and hu according to Eqs. (18), (19) and (22); 3. Compute IðuÞ according to Eq. (27). Here, the time complexity of CAEA is OðnÞ and n is the total number of nodes in the WMN. 6. Scheduling

u

2

Once the optimal bandwidth allocation fgðuÞ; u 2 V 2 g has been obtained by solving the LPs we formulated in Section 4, the next step is to generate a schedule that would achieve the corresponding rate requirements. Besides, the algorithm should be able to minimize the total schedule time. The difficult part of the optimization problem is how to select the simultaneous transmissions in each time slot so as to minimize the total schedule length. Here, we introduce two scheduling strategies:

Step 3: Compute IðuÞ; 8u 2 V 10: for each u 2 V 11: sum 0 12: if CHIðuÞ 6¼ / P 13: sum v2CHIðuÞ[INT ðuÞ ðgv  1 þ hv Þ 14: IðuÞ maxðdsume; gu Þ

&

X

IðuÞ ¼ max gu ;

’! ðgv  1 þ hv Þ

;

8u 2 V

v2CHIðuÞ[INT ðuÞ

ð27Þ (

0 n o ðu 2 V 1 Þ P 1 min gj 16i6g 2 Cðu;iÞ  f ðuÞ P 0 ðu 2 V 2 Þ P 1Cðu;iÞ f ðuÞ 16i6gu 1 2 ; u2V2 and hu ¼ 1Cðv;g Þ where

gu ¼

2

u

Here, we take the example in Fig. 2 to illustrate our optimal solution of the element assignment. We assume every stream has a same capacity of 54 Mbps. In Fig. 2, each node is assigned with three elements. Due to the interference, the system can transmit at most in three independent streams simultaneously. When taking the half-duplex feature into consideration, we can get the maximum guaranteed bandwidth allocation fgðuÞ ¼ 27 Mbps;u 2 fa; c; egg. If we use this bandwidth allocation as an input, we can calculate the optimal element assignment as fIðuÞ ¼ 1; u 2 fa; c; e; f g; IðbÞ ¼ 3; IðdÞ ¼ 2g, which saves 50% cost on the total elements. We list the details in Table 3.

1. Traffic-aware: schedule links with heavy traffic load first. 2. Stream-controlled: (1) use the streams with large capacity first; (2) avoid launching too many streams from the same node at a time slot for the purpose of alleviating interference on other receivers. We propose the Traffic-aware Stream-controlled Link Scheduling (TSLS) algorithm as shown in Table 5. The time complexity is Oðn4 Þ, and n is the total number of nodes in the WMN. In each time slot, we greedily select as many simultaneous transmissions as possible. The principle of the link selecting is based on the two scheduling strategies. Considering that some bottleneck nodes bear heavy aggregate traffic, we give the first strategy higher priority when the two strategies conflict. When more than one link is scheduled in a time slot, following conditions should be met: 1. Send/receive constraint: A node cannot send and receive in the same time slot.

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Table 5 Algorithm 2. Traffic-aware Stream-controlled Link Scheduling (TSLS)

7. Discussion

Input: Network topology graph GðV ; EÞ, Gateway nodes V 1 , Non-gateway nodes V 2 , Element assignment fIðuÞ; 8u 2 V g, Bandwidth allocation fgðuÞ; 8u 2 V 2 g, Traffic flow ff ðuÞ; 8u 2 V 2 g, Stream capacity fCðu; iÞ; 8u 2 V 2 ; 1 6 i 6 kg, Number of time slots per second: M Output: Scheduling length S, Schedulable bandwidth allocation fgðuÞ  MS ; 8u 2 V 2 g Main Program: 1: S 0 2: 8u 2 V , trafficðuÞ f ðuÞ 3: do while 9u 2 V 2 ; trafficðuÞ > 0 4: S Sþ1 5: RENEW WMNðÞ 6: do while ðu SELECT NODEðÞÞ 6¼ 1 7: SCHEDULEðuÞ Subroutine RENEW_WMN(): 8: for each u 2 V 9: resourceðuÞ IðuÞ, uselinkðuÞ 0 10: if ðhopðuÞ  SÞmod2 ¼ 0 11: statusðuÞ send 12: else statusðuÞ receive Subroutine SELECT_NODE(): 13: P /, maxtraffic 0 14: for each u 2 V 2 15: if statusðuÞ ¼ send and resourceðuÞ > 0 16: if trafficðuÞ > maxtraffic 17: P fug, maxtraffic trafficðuÞ 18: else if trafficðuÞ ¼ maxtraffic, P P [ fug 19: if P ¼ /, return 1 20: else return u, such that uselinkðuÞ ¼ minfuselinkðvÞ; 8v 2 P g Subroutine SCHEDULE(u): 21: uselinkðuÞ uselinkðuÞ þ 1, 22: resourceðuÞ resourceðuÞ  1 23: trafficðuÞ trafficðuÞ  Cðu;IðuÞresourceðuÞÞ M 24: for each v 2 NEIðuÞ 25: if statusðvÞ ¼ receive and CHIðvÞ 6¼ / 26: resourceðvÞ resourceðvÞ  1 27: if resourceðvÞ ¼ 0 28: 8w 2 NEIðvÞ 29: if statusðwÞ ¼ send, resourceðwÞ 0

In this section, we discuss the implementation issues of the proposed algorithm framework for the bandwidth allocation and element assignment problem in the WMNs with MIMO links. In Section 4, we formulate the LP solutions for the optimal bandwidth allocation. It provides a tight upper bound for max–min or proportional fairness model. However, as the number of nodes in the mesh network increases, the cost to compute the optimal solutions by LP will increase dramatically due to the inherent nature of the problem complexity. Hence, the LP solutions is only suitable for off-line decisions for building or analyzing the static WMN architecture. If the system would have to recalculate the allocation and assignment strategy on-line, according to the topology change or new traffic requirement in a large WMN, finding the optimal solution is not practical. In Sections 5 and 6, we also proposed efficient heuristic algorithms (TSLS, CAEA) to solve the on-line bandwidth allocation and element assignment problem. Those algorithms can be used in different kinds of dynamic network change scenario. Here are some examples: (1) nodes fail or new nodes would join the network; (2) some elements in the antenna arrays breakdown or need replacement; (3) network topology changes caused by mobility. Whenever these changes happen, the system needs to compute new feasible element assignment and bandwidth allocation in real-time. Otherwise, the whole network may crash due to the unpredictable traffic and interference, or cannot satisfy the application requirements.

2. Transmission DOF Constraint: A node u can launch at most IðuÞ streams simultaneously. 3. Reception DOF Constraint: When a node u is receiving, all the incoming steams including the interference should not be great than IðuÞ. As we mentioned in Section 4, the TSLS algorithm also provide a heuristic solution for the bandwidth allocation problem, which can be an efficient substitute for the LP solutions. If we consider the max–min fairness model, we use fgðuÞ ¼ a; 8u 2 V 2 g as the bandwidth allocation input where a is an initial value of bandwidth allocation. When we consider the proportional fairness model, we can input fgðuÞ ¼ reqðuÞ; 8u 2 V 2 g. The resulting proportional coefficient is MS and the bandwidth allocation is fgðuÞ ¼ MS reqðuÞ; 8u 2 V 2 g.

8. Simulations In this section, we evaluate the performance of our proposed heuristics in a variety of settings. As mentioned above, to the best of the authors’ knowledge, this is the first algorithm framework in WMN/MIMO that considers the joint problem of bandwidth allocation, element assignment and scheduling. Therefore, no comparison can be made with existing schemes. A C-coded custom simulator is used for the performance evaluation of the proposed algorithms. According to 802.11 protocols, we assume the stream capacity is 54 Mbps. The transmission range is set to 35 m and the interference range is set to 70 m. Also, we suppose there is no interference from external networks. We evaluate our algorithms in three scenarios with 1, 2, and 4 gateways, respectively; the positions of the gateways are fixed and pre-located as shown in Fig. 3. Two types of network topologies are studied in our experiment: grids and random topologies. For the grid topology, we use a 5  5 grid, where the total number of nodes including the gateway is 25 and the distance between two adjacent nodes is 35 m. For the random topologies, all network graphs are generated in a terrain of 140 m by 140 m. The number of non-gateway nodes is set from 10 to 120 with an increment of 10. For each test, we generated and ran 50 random con-

J. Wang et al. / Computer Communications 31 (2008) 1372–1384 Gateway node

we use TSLS algorithm to generate a schedulable max guaranteed bandwidth allocation. To alleviate the bias of network topologies, we first investigate the performance under the 5  5 grid network. Fig. 4 indicates the simulation results. For all three subgraphs (a), (b) and (c), the X-axis are the maximum number of elements (k) in each node; the Y-axis are: (a) Maximum Guaranteed Bandwidth for each node with k elements, the data are obtained by solving the LP1 in Section 3; (b) the ratio  of cost saving  by comparing CAEA and EEA, which

Non-gateway node 140M

140M

Transmission Link

1-gateway Scenario

4-gateway Scenario

2-gateway Scenario

b

28

20 16 12 8 4 0 1

2

3

4

5

6

7

 100%: SBA is the

80 70

Cost saving of CAEA (%)

1 gateway 2 gateways 4 gateways

24

GuarBW ðSBAÞ GuarBW ðFBAÞ

schedulable bandwidth allocation from TSLS with CAEA assignment for each node, and the FBA is the optimal bandwidth allocation for each node with k elements. In Fig. 4(a), we demonstrate that the Maximum Guaranteed Bandwidth is linear increasing to the maximum number of elements in a node, as we discussed in Proposition 1. Also, with more gateways, more bandwidth can be allocated to each node because of the alleviation of accumulated traffic. Fig. 4(b) illustrates the cost saving by the comparison of two strategies: CAEA and EEA. We find CAEA can save at least 35% (k ¼ 2) of the cost, and when k > 2, it will save more than 50% on the number of elements. Fig. 4(c) shows the schedulable bandwidth of algorithm TSLS (M ¼ 5000) under the CAEA strategy. Compared with the upper bound by solving the LP prob-

nected graphs and each result shown on the evaluation figures is the mean value. For each specified topology, we investigate the bandwidth allocation problems with two element assignment schemes. The first scheme is to evenly assign each node with k elements (Even Element Assignment – EEA) and the second one is to assign elements according to our CAEA algorithm such that each node is allowed to have a maximum of k elements. In each of these two schemes, we first calculate the max guaranteed bandwidth for each node by solving the LP problem. (In this paper, we use LINGO software [16] to solve the LP problems.) Then,

GuarBW(EEA) (Mbps)

1  costðCAEAÞ  100%; (c) costðEEAÞ

is

Fig. 3. Topology setting.

a

1381

60 50 40 30

1 gateway 2 gateways 4 gateways

20 10 0

8

1

2

k (maximum elements in a node)

3

4

5

6

7

8

k (maximum elements in a node)

GuarBW(CAEA-TSLS) (%)

c 120 100 80 60

1 gateway 2 gateways 4 gateways

40 20 1

2

3

4

5

6

7

8

k (maximum elements in a node) Fig. 4. Performance evaluation of grid topology. (a) Maximum guaranteed bandwidth of EEA. (b) Cost saving of CAEA compared with EEA. (c) Schedulable bandwidth of CAEA compared with maximum guaranteed bandwidth of EEA.

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lem, in most cases, the TSLS can achieve a schedulable bandwidth allocation up to 92% of the optimal. Fig. 5 depicts the performance evaluation for random topologies. In Fig. 5(a), (b) and (c), the X-axis are the number of non-gateway nodes in the wireless mesh network; the Y-axis are (a) Maximum Guaranteed Bandwidthfor each  node with four elements; (b)

1  costðCAEAðk¼4ÞÞ  100%; costðEEAðk¼4ÞÞ

GuarBW ðSBAÞ  100%; k ¼ 4. In Fig. 5(a), we demonstrate (c). GuarBW ðFBAÞ that the Maximum Guaranteed Bandwidth is a reciprocal function of the number of nodes in the network. As the density of nodes increases, the throughput drops dramatically at first (jV 2 j from 10 to 30), then becomes stable when jV 2 j > 70. In Fig. 5(b), we find that as the number of nodes increases the cost saving ratio also increases. CAEA can save at least 48% (jV 2 j 6 30) of the cost, and whenjV 2 j > 30, it will save more than 54% on the number of elements. This is because when the node density increases, the bandwidth for each node decreases; for most nodes, the requirement of elements reduces. Fig. 5(c) shows the result bandwidth of algorithm TSLS (M ¼ 5000) under the CAEA (k ¼ 4) strategy. Compared with the upper bound, the TSLS can achieve a schedulable bandwidth allocation up to 96% of the optimal on average. When we loose the restriction of maximum number of elements per node from 4 to 8, the CAEA (k = 8) scheme outperforms much more compared with the EEA (k ¼ 4).

9. Related work Recent researchers have done great work on the study of capacity for wireless networks [2,17–20,10,5,21]. Gupta and Kumar [2] showed that in a random node placement and communication p pattern ffiffiffiffiffiffiffiffiffiffiffiffiffiwireless network, the per-node throughput is Hð1= n log nÞ, where pffiffiffi n is the number of identical nodes. It becomes Hð1= nÞ under the assumption of optimal node placement and communication pattern. This result suggests that in a very large deployment, the available bandwidth per node may be limited, since it is a decreasing function of the number of nodes in the network. Because the number of simultaneous transmissions will

b 14

1 gateway 2 gateways 4 gateways

12 10 8 6 4 2

0 10 20 30 40 50 60 70 80 90 100 110 120

Number of non-gateway nodes

GuarBW(CAEA-TSLS) (%)

c

Cost Saving of CAEA (k=4) (%)

GuarBW(EEA(k=4)) (Mbps)

a

Fig. 6 illustrates the comparison results in cost saving (Fig. 6(a)) and in throughput improvement (Fig. 6(b)). We find that the CAEA (k ¼ 8) can save the cost by 35% on average when compared with EEA (k ¼ 4); meanwhile, the CAEA (k ¼ 8) can achieve a schedulable maximum guaranteed bandwidth as much as 185% to the optimal solution of EEA (k ¼ 4). Also, as the node density increasing, the CAEA performs better. This result proves our observation that in the wireless mesh networks with some critical nodes bearing both heavy traffic and severe interference, using the same number of elements in each antenna array is not a necessary condition for the improvement of system performance.

62 60 58 56 54 52

1 gateway 2 gateways 4 gateways

50 48

10 20 30 40 50 60 70 80 90 100 110 120

Number of non-gateway nodes

100 99 98 97 96 95 94 93 92 91

1 gateway 2 gateways 4 gateways

90 10 20 30 40 50 60 70 80 90 100 110 120

Number of non-gateway nodes Fig. 5. Performance evaluation of random topologies (k ¼ 4). (a) Maximum guaranteed bandwidth of EEA. (b) Cost saving of CAEA compared with EEA. (c) Schedulable bandwidth of CAEA compared with maximum guaranteed bandwidth of EEA.

J. Wang et al. / Computer Communications 31 (2008) 1372–1384

b

50 45 40 35 30 25 20 15

1 gateway 2 gateways 4 gateways

10 10 20 30 40 50 60 70 80 90 100 110 120

Number of non-gateway nodes

GuarBW(CAEA-TSLS) (%)

Cost saving of CAEA (%)

a

1383

195 190 185 180 175 170 165

1 gateway 2 gateways 4 gateways

10 20 30 40 50 60 70 80 90 100 110 120

Number of non-gateway nodes

Fig. 6. Performance improvement of CAEA (k ¼ 8) compared with EEA (k ¼ 4) under random topologies. (a) Cost saving of CAEA (k ¼ 8) compared with EEA (k ¼ 4). (b) Schedulable bandwidth of CAEA (k ¼ 8), compared with maximum guaranteed bandwidth of EEA (k ¼ 4).

affect the interference dramatically, it is important to relieve the interference so as to improve the capacity of such wireless networks. Multi-Channel Multi-Radio (MCMR) [17–19] and MIMO [20,10,5,21] are two of the new technologies introduced to mitigate interference. The first one is to equip each node with multiple radios that operate on non-overlapping channels so that the nodes can communicate on different channels to avoid the interference. Papers [22,23,12,24] investigate the throughput enhancement for this setting. In paper [22], Raniwala et al. proposed and evaluated one of the first IEEE 802.11-based multi-channel multi-hop wireless mesh network architectures. They developed a set of centralized algorithms for channel assignment, bandwidth allocation, and routing. Alicherry et al. [23] formulated the joint channel assignment and routing problem, taking into account the interference constraints, the number of channels and the number of radios available at each mesh router. They also developed a heuristic algorithm that optimizes the overall network throughput subject to fairness constraints on allocation of scarce wireless capacity among mobile clients. Tang et al. [12] considered a simple max–min fairness model which leads to high throughput solutions with maximum guaranteed bandwidth allocation values and then the Lexicographical Max–Min (LMM) model. Due to the limitation of frequency resources, the requirement for non-overlapping channels has greatly restricted the development of MCMR technique. So MIMO technology emerged with more expectations. The unique features of spatial multiplexing and interference suppression have revealed great potential in the physical layer [7,9,6,11]. And this increased capacity at the physical layer can be exploited to provide improved networking performance by the use of new protocol design techniques [13,25,8,26–28]. In the area of ad hoc networks, there has been some recent interest in understanding how the availability of smart antenna nodes affects both the design and the performance of networking protocols, especially at the medium access control (MAC) and routing layers [25]. In [8], the authors proposed a MAC protocol for ad

hoc networks that leveraged the physical layer capabilities of MIMO links, with the focus being predominantly on the spatial multiplexing capability of MIMO links. The performance optimization for MIMO technique has been study in paper [29–31]. In paper [30], the authors considered the cross-layer throughput optimization problem and formulated a framework that considered data routing at the protocol layer, link scheduling at the MAC layer and stream control at the physical layer. Sundaresan et al. [32] studied the routing problem in Ad hoc networks with MIMO links. They designed different routing protocols for WMN to explore the tradeoff between multiplexing gain and diversity gain. Papers [33–38] studied the scheduling problem. In paper [34], Aniba and Aissa proposed a cross-layer MIMO scheduler to optimize user’s diversity over antennas and to provide high throughput while servicing users in a fairness manner. Anton-Haro et al. [35] discussed packet scheduling in single and multiple antenna wireless systems with QoS support from cross-layer approach. With regards to the optimal element assignment problem, none of the above literatures has mentioned or studied. To the best of our knowledge, we are the first to study the joint problem of bandwidth allocation and element assignment in WMNs with MIMO links. 10. Conclusion and future work In this paper, we first characterize the unique features of MIMO links in wireless mesh networks. Then we define the joint problem of bandwidth allocation, element assignment and scheduling in order to fully exploit the potential advantages of the MIMO technology. We consider both max– min and proportional fair bandwidth allocation schemes, and find the optimal solution using the linear programming method. To take the best advantage of the antenna elements, we propose a Cost-Aware Element Assignment Algorithm to efficiently assign each node with an ondemand basis. In addition, we propose a centralized traffic-aware stream-controlled link scheduling algorithm and

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demonstrate that it provides a schedulable bandwidth allocation which is close to the optimal solution. For future work, we would like to investigate the impact of different routing strategies on the performance of throughput and element assignment in WMN with MIMO support. Acknowledgements This work is supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 113906), CityU Strategic Research Grant (SRG) (Project Nos. 7002217 and 7002102) and the National Basic Research Program of China (973 Program) under Grant No.2006CB303006. References [1] I. Akyildiz, X. Wang, A survey on wireless mesh networks, IEEE Communications Magazine 43 (2005) 23–30. [2] P. Gupta, P.R. Kumar, The capacity of wireless networks, IEEE Transactions on Information Theory 46 (2) (2000) 388–404. [3] D. Gesbert, M. Shafi, D. Shiu, P.J. Smith, A. Naguib, From theory to practice: an overview of MIMO space-time coded wireless systems, IEEE Journal of Selected Areas in Communications 21 (2003) 281–302. [4] A.J. Paulraj, D.A. Gore, R.U. Nabar, H. Bolcskei, An overview of MIMO communications – a key to gigabit wireless, in: Proceedings of IEEE, vol. 92, February 2004, pp. 198–218. [5] D. Shiu, G.J. Foschini, M.J. Gans, J.M. Kahn, Fading correlation and its effect on the capacity of multiple-element antennas, IEEE Transactions on Communications 48 (2000). [6] E. Baccarelli, M. Biagi, C. Pelizzoni, N. Cordeschi, F. Garzia, Interference suppression in MIMO systems for throughput enhancement and error reduction, in: Proceedings of ACM IWCMC, July 2006, pp. 611–616. [7] L. Zheng, D.N.C. Tse, Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels, IEEE Transactions on Information Theory 49 (2003) 1073–1096. [8] K. Sundaresan, R. Sivakumar, M.A. Ingram, T.Y. Chang, A fair medium access control protocol for ad-hoc networks with MIMO links, in: Proceedings of IEEE INFOCOM, vol. 4, 2004, pp. 2559–2570. [9] M.F. Demirkol, M.A. Ingram, Control using capacity constraints for interfering MIMO links, in: Proceedings of IEEE PIMRC, vol. 3, September 2002, pp. 1032–1036. [10] J.B. Anderson, Antenna arrays in mobile communications: gain, diversity, and channel capacity, IEEE Antennas and Propagation Magazine 42 (2000) 12–16. [11] S. Gaur, J.S. Jiang, M.A. Ingram, M.F. Demirkol, Interfering MIMO links with stream control and optimal antenna selection, in: Proceedings of IEEE GLOBECOM, vol. 5, 2004, pp. 3138–3142. [12] J. Tang, G. Xue, W. Zhang, Maximum throughput and fair bandwidth allocation in multi-channel wireless mesh networks, in: Proceedings of IEEE INFOCOM, 2006, pp. 1–10. [13] T. Nandagopal, T.E. Kim, X. Gao, V. Bhargavan, Achieving MAC layer fairness in wireless packet networks, in: Proceedings of ACM MOBICOM, 2000. [14] M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear programming and network flows, third ed., John Wiley & Sons, 2005. [15] L.G. Khachiyan, A polynomial algorithm for linear programming, Doklady Akademiia Nauk SSSR 244 (1979) 1093–1096. [16] Lingo software. Available from: . [17] K. Jain, J. Padhye, V.N. Padmanabhan, L. Qiu, Impact of interference on multi-hop wireless network performance, in: Proceedings of ACM MOBICOM, 2003, pp. 66–80.

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