cross-layer optimization of multichannel multiantenna wmns

Wireless Pers Commun (2013) 71:1443–1459DOI 10.1007/s11277-012-0884-z

Cross-Layer Optimization of Multichannel MultiantennaWMNs

M. Bansal · Aditya Trivedi

Published online: 20 October 2012© Springer Science+Business Media New York 2012

Abstract Use of multiple channels can significantly improve the throughput of wirelessmesh networks (WMNs). Additionally, recent advances in radio technology have made it pos-sible to realize software-defined radio (SDR), which is capable of switching from one channelto another dynamically. On the other hand, equipping wireless nodes with multiple antennascreates great potential for throughput improvement via interference suppression, spatial mul-tiplexing, and spatial division multiple access techniques. In this paper, we investigate the jointoptimization of routing and scheduling in multichannel WMNs, where nodes are equippedwith a single SDR and multiple antenna elements. We analyze achievable throughput of thesenetworks under four different multiantenna modes: single user single stream, single user multistream, multi user single stream, and multi user multi stream, each mode integrates differentcombinations of multiantenna techniques. We mathematically model scheduling and inter-ference constraints and formulate joint routing and scheduling optimization problem withthe objective of maximizing the throughput by minimizing network schedule time such thattraffic demands for a set of sessions are satisfied. A column generation-based decompositionapproach is proposed to solve the problem. Simulation results are presented to evaluate theimpact of number of antennas, number of channels, and number of sessions on the scheduletime for the four proposed modes.

Keywords Multichannel · Multiantenna · Software defined radio · Wireless mesh network ·Routing · Scheduling · Cross-layer design · Column-generation

M. Bansal (B) · Aditya TrivediABV-Indian Institute of Information Technology and Management, Gwalior 474010, Indiae-mail: [email protected]

Aditya Trivedie-mail: [email protected]

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1444 M. Bansal, Aditya Trivedi

1 Introduction

Wireless mesh network (WMN) is a promising networking technology that provides appli-cations such as last-mile broadband wireless Internet access, distributed information sharingand storage, and various multi-media services [1,5]. WMNs consist of two types of nodes:mesh routers (MRs) and mesh clients (MCs). MRs are interconnected via wireless links toform a multi-hop backbone network. Some MRs, called gateways, are connected to the wiredInternet. MRs provide Internet access to the mobile MCs as well as relay information hop byhop using wireless medium. A fundamental issue to be dealt with in WMNs is the capacityreduction due to interference between concurrently active links within interference range ofone another [11]. Additionally, the multi-hop nature of WMNs also degrades the capacity.

In recent years, several radio techniques have been proposed in the literature to increase thethroughput of WMNs. One such technique is to use multiple frequency bands/channels in thenetwork. Unlike a single channel network, a multichannel network allows multiple concurrenttransmissions in a neighborhood as long as they work on different channels. Additionally,recent advances in radio technology have made it possible to realize software-defined radios(SDRs) that can be switched from one frequency band to another dynamically. SDR is arevolutionary technology that is viewed as an enabling technology for dynamic spectrumaccess and sharing. Another promising physical layer technology that has the potential toenhance network capacity is multiantenna or multiple input multiple output (MIMO) technol-ogy. MIMO technology may enhance the throughput of WMNs via interference suppression(IS), spatial multiplexing (SM), and spatial division multiple access. In this work, our focusis the cross-layer optimization of WMNs enabled with these advanced technologies.

Although, there are many works that focus on utilizing multiple antennas in single channelWMNs, research on multichannel multiantenna WMNs is very limited and only few resultsare available [13,18]. Applying MIMO technology in multichannel WMNs is challenging.The routing and scheduling interact with each other because activation of different links areinterdependent. In addition, scheduling directly depends upon the interference level, which isdetermined by the resource allocation at the physical layer. Therefore, in order to take advan-tage of MIMO technology in multichannel WMNs, it is necessary to consider a cross-layerapproach to design these networks.

In this paper, we investigate the problem of joint optimization of routing and schedulingin multichannel WMNs, where nodes are equipped with a single SDR and multiple antennaelements. While multiple channels increase throughput by allowing multiple simultaneoustransmission in a common neighborhood, multiple antenna elements exploit spatial dimen-sions to improve the network throughput not only via IS but also via SM, and/or SDMA.We propose four MIMO modes, each mode exploits one and/or more MIMO capabilities toincrease the network throughput. We model the medium access constraints under these modes.The joint routing and scheduling problem falls in to a mixed integer linear programming(MILP) problem, which is very difficult to solve in general. The complexity lies in findinga set of all possible transmission configurations (TCs). The number of TCs increases expo-nentially with the number of links, number of channels, and the number of antennas. In fact,the problem of enumerating all the TCs is a non-deterministic polynomial-time hard prob-lem [25]. Therefore, we employ a column generation (CG) based decomposition approach,which generates TCs on-the-fly rather than a priori. Our objective is to minimize the scheduletime of the network to satisfy traffic demands for a set of end-to-end sessions along withfinding the routing and scheduling solutions. The schedule time refers to the time in secondsrequired by the network to deliver the traffic demands of all the sessions. It may be noted thatminimizing the schedule time is equivalent to maximizing network throughput. Simulation

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Cross-Layer Optimization of Multichannel Multiantenna WMNs 1445

results are presented to evaluate the impact of number of antennas and number of channels onthe schedule time of multichannel multiantenna WMNs under the different MIMO modes.

The remainder of this paper is organized as follows. Section 2 presents MIMO backgroundand related work. Network model is presented in Section 3. In Section 4, objectives and pro-posed solution approach are discussed. Simulation results are given in Section 5. Finally,Section 6 concludes the paper.

2 MIMO Background and Related Work

Multiple antennas have recently emerged as a powerful technology to enhance the perfor-mance of wireless networks. Under suitable channel fading conditions, multiple antennasprovide spatial dimensions for communication, which can be used to increase data rate byspatial multiplexing and/or increase spatial reuse by interference suppression.

2.1 Single User Multiple Input Multiple Output (SU-MIMO) Communication

SU-MIMO communication refers to a point-to-point communication scenario in cellular ter-minology in which a transmitter can transmit to only one receiver and a receiver can receivefrom one transmitter at a time.

2.1.1 SU-MIMO Spatial Multiplexing

In rich scattering environments, a MIMO link with KT transmit and K R receive antennascan provide k = min(KT , K R) DOFs. These degrees of freedom (DOFs) can be exploited totransmit k independent data streams over the MIMO link, thus lead to k fold increase in thedata rate. Such SM can be implemented by singular value decomposition (SVD) architecture[21] if the channel state information (CSI) is known to both the transmitter and the receiver,or by vertical bell labs space-time (V-BLAST) architecture [27] if the CSI is known only atthe receiver. A linear zero-forcing (ZF) receiver [23] can receive and detect k data streamsif k ≤ K R .

2.2 Multi User Multiple Input Multiple Output (MU-MIMO) Communication

Multi user multiple input multiple output refers to the two MIMO communication scenariosin cellular terminology: the downlink or broadcast channel case and the uplink or multipleaccess channel case. In the downlink scenario, a node attempts to transmit data to multipleusers and in the uplink scenario a group of users attempts to transmit data to a common node.

2.2.1 MU-MIMO Downlink Spatial Multiplexing

If the CSI is known to both transmitter and receiver, then with a suitably chosen precodingscheme [7,22] at the transmitter, a MU-MIMO downlink spatial multiplexing channel withKT transmit antennas and U users, each equipped with K R antennas provides a multiplexinggain equal to the number of transmit antennas, i.e., KT . KT independent non-interfering datastreams can be transmitted to multiple users. The number of antennas at the receiver upperbounds the number of streams that can be detected (using a linear ZF receiver). In downlinkmulti user spatial multiplexing, DOFs at the receiver will depend on the number of antennasat the receiver.

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In the absence of CSI at the transmitter, the DOFs gain reduces to unity, if each user isequipped with single antenna [14], or min(KT , K R), if each user is equipped with K R anten-nas. In the latter case, the transmitter may decide to either multiplex several data streams toa single receiver or spread the streams over multiple users.

2.2.2 MU-MIMO Uplink Spatial Multiplexing

A MU-MIMO uplink spatial multiplexing channel with a base station and U users eachequipped with KT transmit antennas provides a multiplexing gain equal to the number ofantennas K R at the base station. Each user splits its data and encodes them into independentstreams of information with user u ∈ U employing Ku ≤ min(K R, KT ) streams (just asin the SU-MIMO channel). The base-station uses the linear ZF receiver to decode the datastreams of the users.

2.3 MIMO Interference Suppression

Co-channel interference arises due to frequency reuse in wireless channels. When multipleantennas are used, the differentiation between the spatial signatures of the desired signal andco-channel interfering signals can be exploited to mitigate interference. If CSI is availableonly at the receiver, then the receiver equipped with K R antennas can successfully differ-entiate k data streams using linear ZF or minimum mean square error (MMSE) methods ifk ≤ K R . If CSI is known to both transmitter and receiver, then interference suppression canalso be implemented at the transmitter.

2.4 Related Work

The cross-layer design of single channel MIMO-based WMNs has been considered in manyrecent works. The works in [3,15,19,20,26] were focused on jointly optimizing schedul-ing and resource allocation in multiantenna WMNs. In [26], the authors considered a jointbandwidth allocation, antenna element assignment, and stream control scheduling problemin MIMO-based WMNs and presented a heuristic traffic aware stream control schedulingalgorithm. The authors in [15] formulated a demand based fair stream allocation problem asan integer linear program (ILP) and then solved this ILP in conjunction with binary searchto find minimum length transmission schedules. They also presented a greedy heuristic forstream scheduling. Authors in [19,20] considered a cross-layer optimization problem to findlink scheduling and stream control solution with the objective of minimizing the frame lengthto satisfy the traffic demands of all the links. Approximate algorithms were also developedfor stream control scheduling. In [3], we investigated the joint scheduling and stream allo-cation problem under fairness constraints for single-user communication. The authors in[12] characterized the maximum achievable throughput of MIMO based multi-hop wirelessnetworks. Characterizing optimal throughput of MIMO networks with variable-rate streamcontrol has been investigated in [24]. Liu et al. [17] developed a simple model for MIMOchannel capacity computation at the physical layer and order based interference cancelationmodel at the link layer. These models were applied in cross-layer performance optimizationof a multi-hop network enabled with IS and SM techniques. The works in [2,4,6], con-sidered the joint design of routing, scheduling, and physical layer resource allocation forsingle channel MIMO enabled WMNs. In [4], the authors formulated a joint routing andstream control scheduling problem in WMNs. An LP relaxation of the problem is solved,but the solution is not feasible in terms of schedulability; they further proposed a heuristic

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algorithm that schedules a scaled down version of this solution. In [6,28], a CG method is usedto solve a joint routing and scheduling optimization problem in WMNs employing MIMObeam-forming at physical layer. Formulation in [28] is based on protocol interference modelwhile in [6] formulation is based on physical interference model. Authors in [29] proposed aCG based method to solve a cross-layer optimization problem in WMNs where MRs utilizeIS, SM, and SDMA techniques.

However, the research on how to exploit MIMO to enhance throughput of multichannelWMNs is very limited and only few results are available like in [13,18]. The authors in [18]studied the joint optimization of routing, scheduling, channel allocation, stream control inmultichannel WMNs with MIMO links. An algorithm is presented to jointly determine sched-uling, stream control, and channel allocation. However, no results are presented to evaluatethe performance of the algorithm. In [13], authors derived a framework that characterizes themaximum throughput in multichannel WMNs where MIMO is used to increases the through-put via IS only. They optimized the throughput by only considering routing but they did notconsider scheduling in their formulation. They used LP constraint relaxation technique andan approximation algorithm to solve the problem, which provides the suboptimal solution.Different from work in [13], we not only consider scheduling in our formulation but alsoexploit MIMO for SM and SDMA in addition to IS. In summary, our main contributions areas follows:

– We design and model the medium access constraints for multichannel multiantennaWMNs under the four different MIMO modes.

– A mathematical formulation for joint routing and scheduling optimization problem ispresented and a CG-based decomposition approach is used to solve the problem.

– Schedule time performance of the network under the four MIMO modes is evaluated asa function of number of antennas, number of channels, and number of traffic sessions.

3 Network Model

In this paper, we consider a WMN, which is modeled as directed graph G = {N , L}. The setN = {n, n = 1, 2, . . . , |N |} represents all the MRs and the set L = {l, l = 1, 2, . . . , |L|}represents all directed links. The distance between node m and node n is denoted by d(m, n).A directed link l = (m, n) ∈ L exists between node m to node n if d(m, n) ≤ RT , where RT

is the transmission range. For each link l ∈ L, t (l) and r(l) denote the transmitter and thereceiver, respectively. For every node n ∈ N , we denote by L+

n the set of outgoing links fromnode n, and L−

n denotes the set of incoming links to node n, and Ln = L+n ∪ L−

n denotes theset of links incident on node n.

We assume that there are total B frequency channels in the network. Each node is assumedto be equipped with a single radio having K antenna elements. A node with K antenna ele-ments is said to have K DOFs, which it can use for IS and/or SM and/or SDMA. We assumean open-loop MIMO system where interference suppression via multiple antennas is per-formed only at the receiver side. A receiver utilizes one DOF per data stream received fromits intended transmitter and one DOF per interfering stream that needs to be suppressed.A MIMO link l with K antenna elements at both the transmitter and the receiver can supportk ≤ K spatially multiplexed independent data streams. We denote a communication overlink l working on channel b and transmitting k independent data streams by (l, b, k). Notethat there may be B × K such possible communications over link l ∈ L. However, at anygiven time, at most one of these communications could be active. Let X = {(l, b, k), (l =

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1, 2, . . . , |L|), (b = 1, 2, . . . , B), (k = 1, 2, . . . , K )} denote the set of all the possible com-munications in the network. For each communication (l, b, k) ∈ X , c(l, b, k) denotes thecapacity in bits per second, which is a function of the number of data streams k. In thispaper, we consider four different MIMO modes, each utilizes DOFs to achieve one or moreperformance goals as follows:

− Single-user single-stream (SUSS) mode: In this mode, a transmitting node may trans-mit only one data stream along any one of the outgoing links and a receiver node mayreceive only one data stream from any one of the incoming links using one DOF whilethe remaining (K − 1) DOFs may be used to suppress (K − 1) interfering streams.

− Single-user multi-stream (SUMS) mode: In this mode, a node may transmit up to Kspatially multiplexed data streams along any one of the outgoing links and a receivernode may receive k ≤ K spatially multiplexed independent data streams from any oneof the incoming links.

− Multi-user single-stream (MUSS) mode: In this mode, a node may transmit up to total Kindependent data streams along multiple outgoing links with at most one stream alongany outgoing link and a receiver node may receive and decode k ≤ K independent datastreams from multiple incoming links using k DOFs.

− Multi-user multi-stream (MUMS) mode: In this mode, a node may transmit up to Kindependent data streams along multiple outgoing links with multiple streams along anylink and a receiver node may receive and decode k ≤ K independent data streams onmultiple incoming links using k DOFs.

Note that in SUMS, MUSS, and MUMS modes a receiver node while receiving k ≤ Kindependent data streams using k DOFs may simultaneously suppress (K − k) interferingstreams using the remaining (K −k) DOFs. The SUSS mode utilizes IS capability of MIMO;the SUMS mode exploits IS and/or SM; the MUSS mode uses IS and/or SDMA; and theMUMS mode exploits combination of all the three MIMO capabilities, namely IS, SM, andSDMA to increase the network throughput.

3.1 Medium Access Constraints

We assume that a link can be active on at most one channel at a time because of singleradio constraint. However, a link can be active on two different channels during two differenttime slots. We also assume that a node cannot transmit and receive simultaneously. In thispaper, medium access is based on time-division multiple access (TDMA), where time isdivided into variable length time slots. To model the medium access constraints, we use thenotion of a transmission configuration (TC). A TC is a subset of communications that cansimultaneously be active without violating radio and interference constraints. The radio andinterference constraints are discussed below in this subsection. Let T = {Ti : 1 ≤ i ≤ |T |}denote the set of all the TCs of communication set X . We define each Ti as a 1 × |X | rowvector with its each element Ti (l, b, k) is given as

Ti (l, b, k) ={

1, if communication (l, b, k) is active in Ti

0, otherwise

In each time slot only one of the TCs can be active. Let λi denotes the length of time slot forwhich Ti is active. A schedule S can be defined as the set of pairs {(Ti , λi , i = 1, 2, . . . , |T |}.The schedule time of the schedule S is given by

∑|T |i=1 λi .

Now, we mathematically model the radio and interference constraints on multichannelmultiantenna WMNs under the four MIMO modes.

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1. Radio constraintsSUMS mode: A node cannot transmit and receive simultaneously. We have

∑(l,b,k)∈X :

l∈Ln

Ti (l, b, k) ≤ 1, ∀n ∈ N ,∀Ti ∈ T . (1)

Constraint (1) also ensures that a link could be active on at most one channel at a time.SUSS mode: In this mode, a link can communicate at most one data stream using oneDOF, radio constraint is similar to (1), except we have to fix k = 1, ∀l ∈ L.MUMS mode: In MUMS mode, each node can be active in more than one link at a time.Nonetheless, all of the active links incident to a node have to be either in transmissionmode or reception mode. We have, ∀n ∈ N ,∀Ti ∈ T ,∀l ∈ L−

n ,∀l ′ ∈ L+n∑

b∈B

∑k∈K

Ti (l, b, k) +∑b∈B

∑k∈K

Ti (l′, b, k) ≤ 1. (2)

In addition, since each node is equipped with a single radio, all active outgoing links froma transmitter node must work on the same channel at a time. Similarly, all active incominglinks to a receiver node must work on the same channel. These channel constraints aregiven in (3) and (4).∀n ∈ N ,∀b ∈ B,∀Ti ∈ T ,∀l, l ′ ∈ L+

n∑k∈K

Ti (l, b, k) +∑

b′∈B,b′ �=b,l ′ �=l

∑k∈K

Ti (l′, b′, k) ≤ 1, (3)

∀n ∈ N ,∀b ∈ B,∀Ti ∈ T ,∀l, l ′ ∈ L−n∑

k∈K

Ti (l, b, k) +∑

b′∈B,b′ �=b,l ′ �=l

∑k∈K

Ti (l′, b′, k) ≤ 1, (4)

Recall that under MUMS mode, a node can transmit multiple independent data streamsalong multiple outgoing links. Because the maximum number of streams communi-cated by a node along one or multiple outgoing links should not exceed K , we have thefollowing transmit DOFs constraint:

∑(l,b,k)∈X :

l∈L+n

kTi (l, b, k) ≤ K , ∀n ∈ N ,∀Ti ∈ T . (5)

MUSS mode: For MUSS mode radio constraints are similar to (2)–(5), except we haveto fix k = 1, ∀l ∈ L.

2. Interference constraintsWireless interference in the network is modeled by the protocol model [11]. In thismodel, interference range RI = (1 + α)RT , where 0 ≤ α ≤ 1 is the interferencerange coefficient. An active link l = (m, n) ∈ L interferes with another active linkl ′ = (m′, n′) ∈ L at node n′ if (a) d(m, n′) ≤ RI ; (b) m, n, m′, and n′ are all dis-tinct nodes; and (c) l and l ′ are active on the same channel. For each link l ∈ L, letI(l) = {l ′, d(t (l ′), r(l)) ≤ RI , l �= l ′, l ′ ∈ L} denotes the set of interfering links.As the wireless channel is a shared medium, a transmission can be corrupted by interfer-ence from neighboring nodes. Interference constraint depends upon the techniques being

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used at the physical layer. According to the MIMO’s receiver-side interference suppres-sion technique, a receiving node can successfully receive and decode k independent datastreams from its intended transmitter if the number of interfering streams at the receiverare no more than (K − k). The interference constraint under the four MIMO modes canbe formulated as:SUMS and MUMS modes:∀(l, b) ∈ L × B, and ∀Ti ∈ T

(Ω − K )∑k∈K

Ti (l, b, k) +∑

l ′∈I(l)∪L−r(l)

∑k∈K

kTi (l′, b, k) ≤ Ω (6)

where Ω is an integer equal to the number of possible concurrent streams, i.e., Ω =|L| × K . If link l is active in Ti (i.e.,

∑k∈K Ti (l, b, k) = 1), then (6) ensure that total

number of active links interfering with the reception on link l on channel b does notexceed (K − k). On the other hand, if link l is inactive (i.e.,

∑k∈K Ti (l, b, k) = 0), then

r(l) can be overloaded, in the worse case, by as many as Ω streams because in this caseno interference needs to be suppressed.SUSS and MUSS modes: For SUSS and MUSS modes interference constraints aresimilar to (6), except we have to fix k = 1,∀l ∈ L.

Constraints (2)–(4) are proposed by the authors for multichannel multiantenna WMNs,whereas constraints (1), (5), and (6) have been extended to include multiple channels.

3.2 Flow Balance Constraints

We consider a set F of end-to-end sessions in the network. A session f ∈ F is identified bythe triplet {src( f ), dst ( f ), s( f )}, where src( f ) and dst ( f ) denote the source and destina-tion nodes of session f , respectively and s( f ) represents the traffic demand for each sessionf , which is to be transported from the source node src( f ) to destination node dst ( f ). Letx f (l, b, k) denotes the amount of traffic that is assigned to link l active on channel b andcommunicating k streams by the routing scheme corresponding to session f . Then, for thesource node of each session, the net amount of traffic flow going out should be equal to itstraffic demand, i.e., ∀ f ∈ F

∑(l,b,k)∈X ,

l∈L+src( f )

x f (l, b, k) −∑

(l,b,k)∈X ,

l∈L−src( f )

x f (l, b, k) = s( f ) (7)

Similarly, for the destination node of each session, the net amount of traffic flow going outshould be equal to negative of its traffic demand, i.e., ∀ f ∈ F

∑(l,b,k)∈X ,

l∈L+dst ( f )

x f (l, b, k) −∑

(l,b,k)∈X ,

l∈L−dst ( f )

x f (l, b, k) = −s( f ) (8)

Finally, at an intermediate node for session f , the following flow balance constraint musthold, ∀ f ∈ F and ∀n ∈ N \ {src( f ), dst ( f )}:

∑(l,b,k)∈X ,

l∈L+n

x f (l, b, k) −∑

(l,b,k)∈X ,

l∈L−n

x f (l, b, k) = 0 (9)

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Link capacity constraint: The sum of traffic flows passing over a link cannot exceed the meanlink capacity, i.e., ∀(l, b, k) ∈ X

∑f ∈F

x f (l, b, k) ≤|T |∑i=1

Ti (l, b, k)c(l, b, k)λi . (10)

4 Objective and Proposed Solution Approach

Given a set of end-to-end sessions in the network, each with a traffic demand, our objectiveis to maximize the throughput of the network by minimizing the total schedule time suchthat the traffic demands of all the sessions are satisfied. Mathematically, this problem isformulated as:

P1 : Minimize|T |∑i=1

λi

subject to constraints (7)–(10).

Variables x f (l, b, k) ≥ 0, ∀ f ∈ F and ∀(l, b, k) ∈ Xλi ≥ 0, ∀i = 1, 2, . . . , |T |

(11)

The problem P1 is a linear programming (LP) problem that can be solved easily to obtainminimum schedule time and routing solutions, if all the TCs are known. However, the numberof TCs grows exponentially with the network size, number of antenna elements, and numberof channels. Therefore, enumerating all the TCs is not practical. To obtain the optimal solu-tion without enumerating all the TCs, the problem is decomposed into subproblems using acolumn generation (CG) approach. The CG approach, originally presented in [9,10] and usedin some recent works [16,28], is an optimization technique that decomposes a LP probleminto a restricted master problem (RMP) and a pricing problem (PP). These two subproblemsare described in the following subsections.

4.1 Restricted Master Problem

The RMP is similar to the original problem P1, except it is given a set of initial TCs T ′ ⊆ T .The PP is a TC or column generator that keeps generating new TCs as long as there exist onethat when added to RMP can improve the objective of RMP. The RMP formulation is givenas follows:

Minimize|T ′|∑i=1

λi (12)

subject to constraints (7)–(9),

∑f ∈F

x f (l, b, k) ≤|T ′|∑i=1

Ti (l, b, k)c(l, b, k)λi , ∀(l, b, k) ∈ X . (13)

Variables x f (l, b, k) ≥ 0 ∀ f ∈ F and ∀(l, b, k) ∈ Xλi ≥ 0, ∀i = 1, 2, . . . , |T ′| (14)

First instance of the RMP with set T ′ is solved to obtain the primal solution {x f (l,b, k),∀(l, b, k) ∈ X ; λi , i = 1, 2, . . . , |T ′|}. We also obtain the dual variables {ω(l,

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b, k),∀(l, b, k) ∈ X } corresponding to capacity constraint (13). To find if this solutionis also the optimum to the original problem P1 or not, we examine the reduced cost1 − ∑

(l,b,k)∈X ω(l, b, k)Ti (l, b, k)c(l, b, k). If all the TCs Ti ∈ T but not in T ′ have anon-negative reduced cost, then the current solution to the RMP is the optimum solution tothe original problem. On the other hand, if reduced cost is negative, the current solution tothe RMP is not optimum solution to the original problem, and a new TC (i.e., column) withnegative reduced cost needs to be generated to improve the objective of the RMP.

In the CG method, instead of computing the reduced costs for all the columns Ti ∈ T butnot in T ′, we consider the problem of minimizing the reduced cost subjected to the constraint∑

(l,b,k)∈X ω(l, b, k)Ti (l, b, k)c(l, b, k) > 1 and the medium access constraints for differentmodes described in Section 3.1. This optimization problem is called the pricing problem(PP). The PP formulation is presented in the next subsection.

4.2 Pricing Problem

The objective of the pricing problem (PP) is to find a TC that maximizes∑

(l,b,k)∈Xω(l, b, k)Ti (l, b, k)c(l, b, k), which is equivalent to minimizing the reduced cost. The PPformulation is given as follows:

Maximize∑

(l,b,k)∈Xω(l, b, k)Ti (l, b, k)c(l, b, k) (15)

subject to constraint:{(1) and (6), if SUMS or SUSS mode is considered.(2)–(6), if MUMS or MUSS mode is considered.

Varables Ti (l, b, k) ∈ {0, 1}, ∀(l, b, k) ∈ X (16)

If the objective value of the PP is greater than 1, then the column generated by the PP ispassed to the RMPs and the next iteration starts. If the objective value of PP is less than orequal to 1, this means the solution {x f (l, b, k),∀(l, b, k) ∈ X and λi , i = 1, 2, . . . , |T ′|} ofRMP in the current iteration is the optimal solution to the original problem and the columngeneration procedure terminates.

A proof of optimality of solution by CG approach is given in [8]. In [2], we have shownby simulations that the column generation procedure converges to the optimum.

5 Simulation Results

5.1 Simulation Setup

In this section, simulation results are presented to evaluate the performance of the four MIMOmodes with respect to number of antennas, number of channels, and number of traffic ses-sions. The transmission range, RT of each node is set equal to 250 m. The interference rangeis set equal to 400 m corresponding to interference range coefficient α = 0.6. We assumethat the link capacity on each channel is same. The capacity for communication (l, b, k) ∈ Xis set equal to σ × ∑k

i=1(1 − 0.1[i − 1]) Mbps, where k is an integer in [1, 5] and σ is asmall random number uniformly distributed in [0.5, 1] [19]. The CG model of the problemhas been implemented and solved using LINGO 12. LINGO is a comprehensive commercialtool to build and solve mathematical optimization problems easily and efficiently. It has apowerful language for expressing optimization problems, a full-featured environment for

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Table 1 Set of end-to-endsessions for Example 1

Session Source node → Destination node

f1 1 → 10

f2 2 → 3

f3 7 → 9

f4 6 → 8

f5 4 → 6

(a) (b)

(c) (d)

Fig. 1 Impact of number of antennas on the schedule time for the different MIMO modes

building and editing problems, and a set of fast built-in solvers capable of efficiently solvinglinear, nonlinear, integer, and several other classes of optimization problems.

Example 1 In this example, 10 WMN instances are randomly generated. In each instance, 10nodes are randomly placed in a 600 × 600 m region and 5 end-to-end sessions, each havinga traffic demand of 20 Mb are generated. The source and destinations nodes of each sessionare given in Table 1. Figures. 1 and 2 show the schedule time as function of number ofantennas and number of channels, respectively. The presented results are the average of themeasurements for all the 10 instances. Note that, the schedule time represents the minimumtime for which network will be active to satisfy the traffic demands of all the sessions and itcorresponds to the maximum throughput of the network.

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(a) (b)

(c) (d)

Fig. 2 Impact of number of channels on the schedule time for the different MIMO modes

Impact of Number of Antennas In the SUSS mode (Fig. 1a), it may be noted that as thenumber of antennas increases, the schedule time first decreases and then becomes constant.For example, consider the case of B = 1. Increasing the number of antennas from 1 to 4decreases the schedule time from 255 to 107 s, whereas it remains constant at 107 s for num-ber of antennas more than 4. Similar behavior is observed for the case of B = 2, however,the schedule time saturates for K = 2. For the case of 3 or more channels, the schedule timesaturates for K = 1. Reason for this behavior is as follows: in the SUSS mode, a multiantennaequipped transmitter node is allowed to transmit only one data stream using only one DOF,whereas a receiver node equipped with multiple antennas use one DOF to receive one datastream and remaining DOFs can be used to suppress interference from nearby interferingtransmitters. Therefore, more antennas a node have, the more nearby interferers’ signals itcan suppress allowing more communications to be active simultaneously. Consequently, itrequires less schedule time to satisfy a given traffic requirement. Since, for a given network,number of channels, and interference range, each node has a fixed number of interferingsignals, increasing the number of antennas beyond that fixed number plus one cannot reducethe schedule time further.

In the SUSM mode (Fig. 1b), for a given number of channels, the schedule time decreasescontinuously as the number of antennas increases. It may be recalled that the SUMS modedecreases the schedule time by exploiting a combination of IS and SM capabilities of multian-tenna system. Thus, when multiple antennas can no longer be utilized to reduce the schedule

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(a) (b)

(c)

Fig. 3 Comparison of the different MIMO modes

time via IS, the SUMS mode spatially multiplexes data streams to increase capacity of alreadyactive links and thereby reduces the schedule time.

The schedule time behavior of the MUSS mode is similar to that of the SUSS mode(Fig. 1c), except the schedule time saturates for more number of antennas than that in theSUSS mode for a given number of channels. For example, with B = 2 the schedule timesaturates for K = 2 in the SUSS modes, whereas it saturates for K = 4 in the MUSS mode.Note that the MUSS mode reduces the schedule time via IS and/or SDMA. Therefore, whenDOFs are not being used for interference suppression, the MUSS mode still reduces theschedule time by allowing the nodes to transmit or receive independent data streams alongmultiple links simultaneously.

Similar to the SUMS mode, under the MUMS mode the schedule time decreases con-tinuously as the number of antennas increases. The reason it that, in addition to exploitmultiple antennas for IS and/or SM, the MUMS mode further decreases the schedule timeby exploiting multiple antennas for SDMA.

Impact of Number of Channels The schedule time behavior as a function of number ofchannels for the different MIMO modes is shown in Fig. 2. It may be noted that, for all theMIMO modes and for a given number of antennas, the schedule time first decreases andthen becomes constant as the number of channels increases. Thus, it may be concluded that,given a network topology and number of antennas, there is an optimum number of channels

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(a) (b)

(c) (d)

Fig. 4 Impact of number of traffic sessions on the schedule time for the different MIMO modes. Number ofchannels is set equal to 2

beyond which multiple channels can no longer reduce the schedule time. Multiple channelsdecreases the schedule time by allowing multiple communications to be active simultaneouslyin the same vicinity. Due to availability of multiple channels in the network, nodes can beactive on idle channels allowing them to avoid interference with nearby signals. Therefore,more channels a network have, the more nearby interfering transmitters’ signals a node cansuppress and hence achieve the lower schedule time. However, for a given topology eachnode has a fixed number of interfering transmitters and if each of these transmitter work onan independent channel, interference in the network is completely suppressed and additionalchannels will not decrease the schedule time further. In our example, 3 channels are sufficientto completely suppress the interference in the network.

Comparison of the MIMO modes In this section, we compare the different MIMO modes.Figure 3 shows the schedule time for the four MIMO modes as a function of number ofantennas for B = 1, 2, and 3 or more channels. Each point is the average of the measure-ments for all the 10 instances. For a given number of channels, all the MIMO modes have thesame network schedule time if each node is equipped with a single antenna. For the numberof antennas greater than one, all MIMO modes provide better schedule time performancecompared to the single antenna case, regardless of number of channels. For example, if eachnode is equipped with four antennas and three channels are available, the network scheduletime decreases by 58, 79, 87, and 88 % in the SUSS, MUSS, SUMS, and MUMS modes

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respectively, compared to the single-channel single-antenna case. It may be noted that theSUMS mode outperforms the SUSS mode because the SUMS utilize both IS and/or SMwhile the SUSS mode utilizes only IS. Similarly, the MUMS mode outperforms the MUSSmode; the reason is that the MUMS mode uses SM in addition to IS and SDMA, which arealso used by the MUSS mode. It may be noted that irrespective of the number of channels,the SUSS mode has worst and the MUMS mode has best performance among all the MIMOmodes if each node is equipped with more than one antennas. It is closely followed by theperformance of the SUMS mode.

Example 2 In this example, we study the the impact of number of traffic sessions on theperformance of the different modes. Simulations are performed on a square grid networkconsisting of 9 nodes distributed over a 500 × 500 m area. Settings of other parameters havebeen kept as above. The schedule times of the network for 15, 10, and 5 random sessions,each session having a demand of 20 Mbs, in the SUSS, SUMS, MUSS, and MUMS modesare shown in Fig. 4a–d, respectively. It may be seen from Fig. 4 that in all the modes, byincreasing the number of sessions, the schedule time of the network is increased.

6 Conclusions

In this paper, we investigated the cross-layer optimization of routing and scheduling in mul-tichannel multiantenna WMNs. We considered the use of multiple channels to improve thenetwork throughput of MIMO-based WMNs. We mathematically modeled the schedulingand interference constraints on the multi-channel multi-antenna WMNs under the four dif-ferent MIMO modes. Our objective was to maximize the network throughput by minimizingthe schedule time of the network such that the traffic demands for a set of end-to-end ses-sions are satisfied. A CG approach is proposed to solve the optimization problem. Simulationresults have been presented to evaluate the network performance. It is shown that multipleantennas reduces the network schedule time in all the MIMO modes and the MUMS modeoutperforms all the other modes. It is also shown that, for a given network topology, whenthe MRs are equipped with a single SDR and multiple antennas, there is an optimum numberof channels beyond which multiple channels can no longer reduce the schedule time.

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Author Biographies

M. Bansal is a Ph.D. student in the Department of Information andCommunication Technology at the ABV-Indian Institute of Informa-tion technology and Management Gwalior, India. He received his B.E.(Electronics) and M.E. (Computer, Communication, and Networks)degrees from the Rajiv Gandhi Technological University in 2001 and2006, respectively. His research interests are in the area of cross layerdesign of wireless mesh networks, with focus on utilizing advancedphysical layer technologies such as MIMO and multi-radio multi-chan-nel in these networks.

Aditya Trivedi is a Professor in the ICT Department of ABV-Indian Institute of Information Technology and Management, Gwalior,India. He received his bachelor degree (with distinction) in Electron-ics Engg. from the Jiwaji University. He did his M.Tech. (Commu-nication Systems) from Indian Institute of Technology (IIT), Kanpur.He obtained his doctorate (Ph.D.) from IIT Roorkee in the areaof Wireless Communication Engineering. His teaching and researchinterest include Digital communication, CDMA systems, Signal pro-cessing, and Networking. He is a fellow of the Institution of Elec-tronics and Telecommunication Engineers (IETE) and a member ofInstitution of Electrical and Electronics Engineers (IEEE), USA.Dr. Trivedi has guided one Ph.D. thesis and presently guiding fivePh.D. theses. He has guided more than fifty M.Tech. theses. Dr. Trivediis a reviewer of reputed IEEE and Springer journals. He has publishedmore than 50 papers in various prestigious International/National jour-nals and conferences. In 2007, he was given the IETE’s K.S. KrishnanMemorial Award for best system oriented paper. He has delivered talksin various places related to wireless communication and networking.

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cross-layer optimization of multichannel multiantenna wmns

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