Capacity of Hybrid Wireless Mesh

8/13/2019 Capacity of Hybrid Wireless Mesh

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Capacity of Hybrid Wireless MeshNetworks with Random APs

Weihuang Fu, Member, IEEE, and Dharma P. Agrawal, Fellow, IEEE

AbstractIn conventional Wireless Mesh Networks (WMNs), multihop relays are performed in the backbone comprising of

interconnected Mesh Routers (MRs) and this causes capacity degradation. This paper proposes a hybrid WMN architecture that the

backbone is able to utilize random connections to Access Points (APs) of Wireless Local Area Network (WLAN). In such a proposed

hierarchal architecture, capacity enhancement can be achieved by letting the traffic take advantage of the wired connections through

APs. Theoretical analysis has been conducted for the asymptotic capacity of three-tier hybrid WMN, where per-MR capacity in the

backbone is first derived and per-MC capacity is then obtained. Besides related to the number of MR cells as a conventional WMN, the

analytical results reveal that the asymptotic capacity of a hybrid WMN is also strongly affected by the number of cells having AP

connections, the ratio of access link bandwidth to backbone link bandwidth, etc. Appropriate configuration of the network can drastically

improve the network capacity in our proposed network architecture. It also shows that the traffic balance among the MRs with AP

access is very important to have a tighter asymptotic capacity bound. The results and conclusions justify the perspective of having such

a hybrid WMN utilizing widely deployed WLANs.

Index TermsAccess points, asymptotic capacity, WLAN, wireless mesh networks, mesh routers

1 INTRODUCTION

WIRELESS Mesh Network (WMN) [1] is emerging as apromising technology in providing ubiquitous high-speed service for Mobile Clients (MCs), also called meshclients. Mesh routers (MRs) play an essential role in aWMN, which provides service for MCs on one hand;forward data packets via wireless link to neighboring MRson the other hand. Interconnected MRs form the backbone

of a WMN, where several special MRs connecting to theInternet with wired cables are called Internet Gateways(IGWs). By taking advantage of wireless multihop forward-ing [2], deployment of MRs poses much less constraints asthey can be deployed on electric poles or house rooftop.Such deployment feasibility enables a WMN to provide lowcost metro-scale coverage for MCs access.

Themajor challengein a WMN is thecapacity degradationproblem caused by the interference on a single or multiplerouting paths during multihop transmission. Although thenetwork architecture of any WMN is different from an ad hocnetwork, the asymptotic capacity bound derived by theanalytical work in [3] is still valid for a WMN backbone. Per-

MR capacity of a randomly deployed backbone with ad hocrouting can be given by

WffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNRlogNR

p

;

where W denotes the maximum backbone link datatransmission rate between a pair of neighboring MRs andNRdenotes the number of MRs. It is obvious that the size of

the network is largely constrained by requirements of per-MR capacity. By optimizing the locations of MRs, per-MRcapacitycanbeimprovedbyafactorof ffiffiffiffiffiffiffiffiffiffiffiffiffiffilogNRp . Actually,MRs need not have access to A/C power as energy can besupplied from self-equipped solar panels. MRs can even bedropped anywhere required. Then, per-MR asymptoticcapacity can be said to approach WffiffiffiffiffiNRp .

By deploying IGWs in the network, the whole WMNforms multiple clusters where each cluster is led by an IGWand constraints MRs closer to the IGW. Readers interested invarious cluster construction methods are suggested to referto [4]. After IGW clusters are formed, the traffic between theMRs in different clusters, i.e., intercluster traffic, are directedto their associated IGWs and utilize the wired connectionsbetween the IGWs. A similar network architecture is thehybrid ad hoc networks, where infrastructures are inter-connected with wired cables and deliver data packets for adhoc clients in a single or multiple hops, as shown in Fig. 1a.The capacity of such an ad hoc network with infrastructure

has been investigated in [5], [6], and [7]. Due to randomdeployment, connectivity has a major impact on theperformance. Two geometrically closeby MCs may have avery long routing path due to weak network connectivity.Recent results [6] indicate that per-MC capacity understrong connectivity can achieve W0

logNC, whereNCdenotes

the number of MCs and W0 denotes the total bandwidth.BandwidthW0is shared by all MCs for ad hoc connection orthe connection to infrastructure with time division multipleaccess (TDMA) scheduling.

Different from an ad hoc network, a WMN is possible toreduce the random effect and improve the network capacity

by appropriately deploying MRs. The effect of the numberof IGWs in a WMN is discussed in [8], where MCs areconnected to MRs in a single hop. When the number ofarbitrary deployed IGWs is ! ffiffiffiffiffiffiffiNRp , the asymptotic

136 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 12, NO. 1, JANUARY 2013

. The authors are with the Department of Computer Science, University ofCincinnati, Cincinnati, OH 45221. E-mail: {fuwg, dpa}@cs.uc.edu.

Manuscript received 23 July 2009; revised 6 May 2010; accepted 4 Nov. 2011;published online 15 Nov. 2011.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2009-07-0299.Digital Object Identifier no. 10.1109/TMC.2011.244.

1536-1233/13/$31.00 2013 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS


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capacity increases with the number of IGWs, as long as datais denominated by the intercluster traffic through the IGWs.If the network capacity needs to fully explore IGWs,adequate number of IGWs need to be deployed, whichposes restrictions on the deployment of MRs. When

the number of IGWs is O ffiffiffiffiffiffiffiNRp , MRs in a cluster with asingle IGW have to be designed to have information aboutthe complete network topology so as to perform routingfunctions without involving the IGW. Otherwise, thecapacity is constrained by the IGWs and the capacity maybe worse than that of ad hoc routing in the backbone.Having the whole network information at the MRs may leadto excessive overhead and may not facilitate easy manage-ment of the clustering approach.

It is noted that MRs, MCs, and IGWs in [8] share the samespectrum with TDMA scheduling, which are different froma common two-tier WMN. The role played by MR in [8] is

more like a relay that has additional forwarding function toMC. It has been pointed out in [9] that current IEEE 802.11MAC protocol cannot achieve a reasonable throughput asthe number of hops increases to four or higher. A two-tierWMN illustrated in Fig. 1b employs dedicated spectrum forthe backbone and different spectrum for access link betweenMR and MC. MCs can use IEEE 802.11, which has beenwidely adopted for Wireless Local Area Network (WLAN)connection. As different notations used in Fig. 1b forbackbone links and access links, MRs can have two typesof wireless interfaces and use multichannel multiradio [10],[11] for backbone connections.

The analysis in the paper will show that the asymptoticcapacity can be further enhanced with our proposed hybridWMN architecture, wherein WMN is augmented by thepresence of wired Access Points (APs). WLAN technology ispopular to provide network access for clients. However,limited coverage can only support relatively small region,like area within a house or an office. Extending the coverageby multiple co-deployed APs requires availability of wiredcables at the AP locations. Second, deploying multiple APsin a large coverage area leads to serious inter-WLANinterference and throughput degradation [2], [12]. Inaddition, the Medium Access Control (MAC) layer of APscannot support handover for MCs. Although many APs can

be deployed, it is still not feasible to provide seamlesscoverage in a city by APs without huge investment on cablesand it is not efficient for APs to provide service directly toMCs. The use of APs in WMN needs to be explored.

In this paper, we propose to have a higher capacitybound for a WMN with three-tier hybrid network archi-

tecture by exploring random AP connections, where MR isallowed to connect to the APs in its coverage. The proposednetwork architecture is illustrated in Fig. 2. Backbone linksuse a dedicated spectrum with bandwidthWbps. MRs andIGWs employ multiple orthogonal channels to isolateinterference regions. Each MR is equipped with multipleradios that are able to operate on different channelssimultaneously. While two neighboring MRs employ thesame channel, a link is formed between them and such alink is called a backbone link. MRs also provide access tothe MCs around its neighborhood. The total data rateavailable for the network access is Bbps, with a dedicated

wireless interface for the access link. APs operate on accesslink spectrum, which shares bandwidth Bwith MCs. Theimpact on capacity from APs is related to the deployment.MRs and IGWs are predeployed by Internet ServiceProvider (ISP) at planned locations for constructing aWMN backbone. In contrast, APs are randomly deployedby users and the connection between AP and MR israndom. APs could be turned off by the users and newAPs could appear to be active. The number of APs could bechanging over time. We take the random deployment ofAPs into account. As we can see from Fig. 1b, two-tierWMN connects to the Internet through a backbone link toIGWs. On the other hand, Fig. 2 shows three-tier WMN to

have a random connection to the Internet through accesslinks via APs. In a later analysis, we show that the ratio ofBto Whas a very important effect on the network capacity,besides the number of APs.

Random deployment, random connection, and usingaccess link spectrum are the special attributes of employingunsystematic APs as an additional third tier for a WMN.There are significant advantages of having such connec-tions. The number of hops that traffic transfers in thebackbone is reduced by utilizing the wired connection ofAPs. As a result, end-to-end delay is also decreased. Thenetwork coverage can also be enhanced as MRs have goodoutdoor coverage while APs have better indoor coverage.

Knowing that WLANs and WMNs are different networks,our scheme provides cooperation between WMN andWLANs by utilizing the residual capacity of WLANs.The focus of different environment enables two types of

FU AND AGRAWAL: CAPACITY OF HYBRID WIRELESS MESH NETWORKS WITH RANDOM APS 137

Fig. 2. Proposed three-tier hybrid WMN with random AP connections.

Fig. 1. Network architecture.


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networks to work together. This three-tier WMN achievescapacity enhancement at near no additional cost asWLANs are already deployed. It is compatible withcurrent wireless network technology and facilitates currentMCs to explore that.

Our main contributions are:

. Proposing a novel network architecturea hybrid

WMN, which will have a higher capacity than aconventional WMN. Presence of APs in the deploy-ment area of conventional WMNs is assumed andaccess links for the connections to MRs are used.Thus, from the network architecture view, existingWMN is extended with new elementsAPs and anew link typeAP-MR link, and translates a two-tiernetwork into a three-tier network;

. Applying analytical model to the proposed networkarchitecture;

. Deriving an asymptotic capacity value for MRs andMCs under various conditions; and

. Analyzing the impact of the number of MR withrandomly located AP access, ratio of backbone linkbandwidth to access link bandwidth, etc.

The rest of this paper is organized as follows: The systemmodel is presented in Section 2. Section 3 performs the per-MR capacity with various traffics, and Section 4 analyzesthe per-MC capacity under grid deployed MRs and randomAP access. The effects of network elements and comparisonwith other networks are discussed in Section 5. Finally, thepaper is concluded in Section 6.

2 NETWORKMODEL AND ANALYSIS

The backbone of a WMN consists of MRs and IGWs,where the MRs and the IGWs are wireless interconnectedto each other and provide service to the MCs. MultipleIGWs divide a WMN into several clusters such that eachone is led by an IGW. We investigate such an IGW clustershown in Fig. 2. MRs in the cluster is homogeneous thathave the same backbone and access link transmissionregion. Due to interference among neighboring MRs, theyhave to share wireless resources in the frequency domainand/or time domain.

If an AP is present in the transmission range of an MRcell,MR can access the AP, as shown in Fig. 2. The connection

between the MR and the AP is also an access link, whichshares bandwidth with the MCs it is serving. We call the MRwith AP connection as AP-MR and the MR without APconnection as non-AP-MR. While we use term MR

without specifying this, and it means both kinds of MRs.The number of AP-MRs is denoted by NA. The locations ofNA AP-MRs in NR MRs are assumed to be uniformlydistributed. APs and IGWs have wired connection to theInternet as shown in Fig. 2. The bandwidth of wiredconnection to APs and IGWs is assumed to be large enoughso as not to pose any constraint on the network capacity.

Fig. 3 shows two examples for random AP connections,which is a portion of the network shown in Fig. 2. There aretwo types of routings that can benefit from the random APconnections. While the traffic is between two MRs, therouting in a conventional WMN is relayed by MRs in thebackbone, denoted by the dotted line in Fig. 3a. Withthe inclusion of APs, it can also select the routing path goingthrough the wired Internet, denoted by the dashed line in

Fig. 3a. By such an alternative routing path, the traffic at MRsin backbone can be reduced. It also diminishes the inter-ference in the backbone. As a result, the capacity can beenhanced. Similarly, when the traffic is between MR andIGW, the alternative path can go through the AP and thewired connection to the IGW, denoted by the dashed line inFig. 3b. Theconventional routing path, denoted by the dottedline in the figure, could easily cause congestion at the IGW.

2.1 Hybrid WMN Architecture

The network is modeled as shown in Fig. 4, where griddeployed MRs are indicated by blue circles, the IGW is

indicated by red rectangles, randomly distributed MCs areindicated by green dots, and APs are indicated by darktriangles. The three tiers of a hybrid WMN are: MCs in thefirst tier (the first column in Fig. 4a) connect to the MRs in thesecond tier. Each MR provides network access service formultiple MCs. MRs in the second tier also interconnect toeach other. So, the traffic from MCs can exchange via theconnections among MRs in the second tier. There is one IGWin the second layer to provide Internet access to MRs. Thethird tier includes APs which are connected to each other bywired connections. The connections from the MRs in thesecond tier to the APs in the third tier are random, which are

denoted by dotted lines in the figure. While wirelessconnections between the second tier and the third tieravailable, the traffic between two MRs in the second tiercan be exchanged via the third tier. It may be noted that the


Fig. 3. Alternate routing with random APs connections.

Fig. 4. Hierarchy and deployment.


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link bandwidth between two nodes in different tiers aredistinct, where the capacity enhancement of hybrid WMNbenefits from the additional tier of alternate Internet access.

Fig. 4b shows the geographic model of our proposedhybrid WMN. MRs are deployed in a unit square denotedbyD. The deployment of MRs follows the grid deploymentmethod, which divides the unit square into NR smallersquares and each square is served by MR located at thecenter of the square. The reason behind a grid deploymentis that the locations of MRs are preplanned and most of thecities such as Manhattan, New York, has grid-based streets,which naturally support the grid deployment. Without lossof generality, NR is the value such that

ffiffiffiffiffiffiffiNR

p is an odd

number. An IGW is deployed at the center square.

The coverage of MR i is a circle Csi; rs centered atpointsiwith radiusrs. Given a region D, MRs should satisfyD Si Csi; rs to seamlessly cover D. To simplify theanalysis, we assume MRs only provide network access

service for MCs within the coverage closer to MRs. In other

words, MRs coverage can be approximated by a smallsquare illustrated in Fig. 4. The size of the small square is1ffiffiffiffiffiNR

p 1ffiffiffiffiffiNR

p . MR is able to connect with APs in its coveragethrough the access link. The total number of APs in D isNA.

MR backbone transmission distance rTshould be able to

reach closeby MR, which is slightly larger than 1ffiffiffiffiffiNR

p . It istrivial to prove that such an MR deployment is connected

with transmission distance larger than 1ffiffiffiffiffiNR

p . To provide fullcoverage to each unit square, MR u needs to provide anaccess service for the small square center at si with at

least side length 1

ffiffiffiffiffiNR

p . Since a circle is generally used todenote the coverage of a cell, we use a circle C

si; rs

containing the square to denote the coverage of MR i, wherers 1ffiffiffiffiffiffiffi2NRp . So, the whole unit square area can be seamlesslyserved by such a backbone network.

With the protocol model given in [3], any transmissionfrom MR i to MR j needs to be within the transmissiondistance rT,jsi sjj rT. To avoid the interference, othertransmitting MRk needs to satisfyjsk sjj rT, whereis a positive constant to ensure a safe geographic gapbetween two simultaneous transmissions. While WMNbackbone employs multichannel multiradio technology forinterference avoidance, the protocol does affect the trans-mission using the same channel.

2.2 Traffic Model

First, traffic generated or terminated at MCs can be dividedinto intercluster and intracluster traffic. The interclustertraffic is from the MCs to the destination outside the clusteror from the source outside the cluster to the MC in thecluster. The traffic goes to the IGW and the IGW is in-chargeof aggregating intercluster traffic, routing in the wirednetwork, protocol conversation, etc. Without loss ofgenerality, we assume inter-cluster traffic is from the MCsto the IGW. This condition also applies in the reversedirection. While random AP access is available in a cluster,

the intercluster traffic can be relayed by the APs to the IGWthrough wired connection or vice versa. We assume that themanagement function is located at the IGW and theintercluster traffic can be exchanged between the AP and

the IGW via the wired connection between them (whichmay go through several routers in the wired Internet).

The other type of traffic is the intracluster traffic, whichis between two MCs in the same cluster. In a WMN, eachMR has the routing ability that can divert the intraclustertraffic. The routing list of the MRs in a cluster can be createdby the same way as in a mobile ad hoc network (MANET)or be periodically configured by the IGW. For intracluster

traffic, there are two ways to route. The traffic can be routedtoward the cluster head (IGW) and then reroute to thedestination MR. This method is simple to implement aseach MR only needs to know the route to the IGW, and theinformation of the MCs cell locations can be simply storedat the IGW. However, since all the traffic goes to or comesfrom the IGW, the capacity is seriously constrained at theIGW. Intracluster traffic with such a routing method showsthe same traffic pattern as that of intercluster traffic. Thecluster forms a tree-like topology that is rooted at the IGW.Network capacity with such a routing method can be easilyderived from the case of the intercluster traffic. Anotherway is to route the intracluster traffic like MANET routingin the form of a mesh topology. Each MR either has thelocation information of the MCs in the cluster or is able toinquire the information from the IGW. The data traffic doesnot have to go through the IGW, so the congestion at theIGW can be mitigated and the capacity of the cluster canbe improved. With MANET type routing among MRs, thecapacity can be improved further by balancing the trafficamong multiple routing paths between the source and thedestination MRs. We assume the MANET routing method isused for intracluster traffic so as to have an enhancednetwork capacity. The asymptotic capacity of the methodthat forwards all traffic to the IGW can be also derived from

our results, which is a special case of the asymptoticcapacity with only intercluster traffic.

2.3 Capacity Definition

In the literature such as [5], [13], the network capacityanalysis is based on the per-node asymptotic capacity ofthe traffic in a single-tier network. We extend thisdefinition to three-tier hybrid WMNs with multiple trafficsdescribed above.

Definition 1 (Asymptotic Capacity). The asymptotic capacityof a node, denoted by NC, is of order gNC bps ifdeterministic constants c >0 and c 0


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into four categories according to the types of devices

involved. The first type is the ad hoc MANET-type traffic

which exists in the backbone only among MRs. The second

one is the random intracluster AP traffic. With wireless

connection between MR and AP, intracluster traffic can

utilize the wired connection between two APs. The third is

the intercluster traffic relayed by MRs through the IGW. And

the fourth one is the random intercluster AP traffic, whichuses the wired connection between the AP and the IGW.

The main results of per-MR capacity are summarized

here. We show per-MR capacity for four different types oftraffic. Then, we determine the per-MR capacity with the

combination of the traffic patterns.Per-MR capacity with ad hoc intracluster traffic which

does not involve AP can be expressed by

Rr Wffiffiffiffiffiffiffi

NRp

: 1

Per-MR capacity with intracluster AP traffic is ex-

pressed by

Rp

NAB

NR

; BOW;

NAW

NR

; B!W:

8>>>>>:

2

From (1) and (2), the combination of two types of traffic

that maximizes the asymptotic capacity is expressed by

RI

W

ffiffiffiffiffiffiffiNRp

; NAO W


pB

;

NAB

NR

; NA! W

ffiffiffiffiffiffiffiNRp

B

^ BOW;

NAW

NR

; NA!


p ^ B!W:

8>>>>>>>>>>>>>>>:

3

Per-MR capacity with intercluster traffic without APs is

expressed by

Rg W

NR

: 4

Per-MR capacity with intercluster AP traffic is ex-

pressed by

Re

NAB

NR

; BOW;

NAW

NR

; B!W:

8>>>>>:

5

From (4) and (5), Per-MR capacity with intercluster

traffic can be expressed by

RE

NAB WNR

; BOW;

NA

1

W

NR

; B!W:

8>>>>>:

6

Per-MR capacity with the composition of all traffic

patterns is expressed by

R

Wffiffiffiffiffiffiffi

NRp

; NAO W


pB

;

NAB

NR

; NA! W


pB

^ BOW;

NA 1W

NR

; NA!


p ^ B!W:

8>>>>>>>>>>>>>>>: 7

3.1 Routing and Traffic Balance

To illustrate the routing and traffic balance scheme used inanalyzing a WMN backbone, we use a 2D grid-based WMNof Fig. 5. Figure also shows the connectivity graph of theMRs, which are denoted by dots at the intersections. Theedges between intersection points denote the communica-tion links. Packets can transmit from one MR to theneighboring MR in the grid, which counts as one-hoptransmission. The location of MRs in the grid can beexpressed by two integers: an integer for the x-axis and the

other one for the y-axis. For example, we have the trafficfrom sourceM R1to destinationM R2. The locations ofM R1andM R2 are denoted byi; jandm; n, respectively. Theminimum number of hops for the transmission is given byh jm ij jn jj. A feasible routing path with theminimum number of hops is illustrated by the blue colorarrows P1 along the grids in Fig. 5.

There are a number of paths that satisfy the shortest pathrouting, which are within the rectangles formed by MR1and MR2 points and are shown in Fig. 5, by the brokenrectangle A1 formed by MR1 and MR2. The traffic can bedistributed and balanced on those paths and is routed bythe set of MRs at the locations within in the rectangular

area (x2 i; m, y2 j; n).Random access to AP enables the traffic to take the

shortcut of the wired connection. There may be anotheralternate routing path. For example, if MR3 and MR4 aretwo MRs with AP access. M R1 can transmit the packets toMR3andM R3sends the packets to its AP(s) with the accesslink. The packets then go through the wired connection(s)between the APs, to the AP(s) at the cell of MR4. It isnoticeable that the number of APs at a cell may be morethan one and the traffic might be distributed to multipleAPs. AfterM R4receives the packets, it relays the packets todestination MR2 via the backbone. The routing path is

denoted by P2 and is illustrated by the red arrows, and thewired connection is denoted by the red dashed line in Fig. 5.

Routing between non-AP-MR and AP-MR can do thebalancing between normal backbone traffic. For example,


Fig. 5. Routing and traffic balance in a WMN backbone.


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the routing paths can take the set of paths within the brokenrectangular area A2 formed by MR1 and MR3, which isillustrated by the broken rectangle between M R1 andM R3in the figure. It is also applicable to MR4 and MR2, asshown by the broken rectangle A3 in Fig. 5. The trafficbalance in the wired connection depends on the configura-tion of the wired network, assuming that the wiredconnections among APs has enough bandwidth and are

able to deliver packets with relatively low latency.Here, some useful lemmas used later are listed. Proofs of

Lemmas 1 and 2 are given in [14] and a proof of Lemma 3 isgiven in [15].

Lemma 1 (Chernoff Upper Tail Bound). If is the sum ofindependent indicator random variables, then

P 1 E e2E2 ; >0: 8

Lemma 2 (Chernoff Lower Tail Bound). If is the sum ofindependent indicator random variables, then

P 1 E e2E2 ; 2 0; 1: 9

Lemma 3.ForNCMCs uniformly and independently distributedin a unit square, the critical transmission distance is

c0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffilogNC

NC

s ; 10

wherec0 is a positive constant greater than one.

3.2 Backbone Ad Hoc Routing

The backbone capacity of intracluster traffic with ad hocrouting, i.e., MR to MR traffic, has higher asymptoticcapacity with a regular grid deployment. Proposition 1provides the asymptotic capacity of MRs with grid deploy-ment which follows the arbitrary network asymptoticcapacity of [3].

Proposition 1. Given a connected backbone network consistingof NR grid deployed MRs, per-MR capacity with ad hoctraffic is Rr WffiffiffiffiffiNRp .

Proof.Traffic with ad hoc routing is between a pair of MRsin the same cluster. With a relative large NR, each MR

has the same probability to be a source or a destination.Two MRs on grid m; n andi; j can be randomlypicked as the source MR and the destination MR,respectively. The number of hops for the routes on thegrids is at least h jm ij jj nj. Considering allsuch possible pairs, the lower bound of the averagenumber of hops hL is computed as follows:

EhL Ejm ij Ejj nj 2EEjm iki: 11Ejm iki is obtained as

E

jm

i

ki

1ffiffiffiffiffiffiffi

NRp Xi1

m1i

m

XffiffiffiffiffiNR

p

mim

i

! i

2ffiffiffiffiffiffiffiNR

p 1ffiffiffiffiffiffiffiNR

p 1

i ffiffiffiffiffiffiffi

NRp 1

2 :

12

Ei2 is computed by following equation:

Ei2 ffiffiffiffiffiffiffi

NRp 12 ffiffiffiffiffiffiffiNRp 1

6 : 13

Substituting Ei2 from (13) into (12), and thensubstituting (12) into (11), we obtain the lower boundof average number of hops

hLEhL 23


p 1ffiffiffiffiffiffiffi

NRp

: 14

Denote the upper bound for the average number ofhops by hU. In any feasible routing scheme illustratedearlier, traffic can be distributed and balanced on theroutes available within the minimum rectangular areaconstructed by the source and the destination MRs. Oneof the attributes of such a routing scheme is that thenumber of hops for different routes is the same and isequal to the number of hops of the shortest path. So, theupper bound of the average number of hops can beobtained by

hUO ffiffiffiffiffiffiffi

NRp

: 15As the maximum data rate is no more than W, the

maximum throughput in the backbone is

RrhNRNRW : 16Due to interference among multiple transmissions, not

all MRs could send data at the same time over the whole

frequency band. For MRs in the interference region, they

have to share the bandwidth by operating at different

frequencies and/or time slots. Given the grid topology of

a WMN backbone, the transmission is set to reach theclosest MRs that does not interfere with another second

closest MRs. We use the channel assignment for the links

between neighboring MRs in the grids so that links in the

same interference region can operate on different ortho-

gonal channels, without interfering with each other. An

interference region is formed by a circle r2I centered at

the transmitting MR. Let us take rT 1ffiffiffiffiffiNRp , then rI 1ffiffiffiffiffiNRp .Each MR occupies a region of 1ffiffiffiffiffi

NRp 1ffiffiffiffiffi

NRp in the unit

square. So, there are at most r2INRMRs in the interference

region, expressed by nIM

r2INR

1

2. The num-

ber of links within the interference region is no more than4nIM 41 2. So, the number of channels requiredis no more than c141 2 for a feasible channelassignment like [16]. With this channel assignment and

scheduling, we can have

RrhNRNRW2c1

: 17

Thus, we obtain the MR ad hoc asymptotic capacityfrom (16) and (17), which is expressed by

Rr Wffiffiffiffiffiffiffi

NRp

: 18tu



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Different from the capacity analysis in a random net-work, some factors can be identified and capacity can beimproved in a regular topology. The average number ofhops in a grid topology is tightly bounded by the optimalrouting scheme and the feasible routing scheme. Thenumber of MRs affected by the interference can also becomputed correctly. As a result, the backbone capacity canbe accurately evaluated taking these factors into account.

The asymptotic capacity in other regular deployment suchas hexagonal topology, triangle topology, and so on, can beestimated with the same method used in Proposition 1.

3.3 Intracluster AP Traffic

For the intracluster traffic between two MCs in the samecluster, it can also utilize random connections between MRsand APs. Decreasing the number of hops for the trafficgoing to the Internet reduces the load in the backbone. Theenhancement is mainly affected by the number of accessibleAPs and the bandwidth of such random connections. Asdescribed later, Propositions 2 and 3 provide per-MRcapacity under different conditions.

Lemma 4.Given a connected backbone network consisting ofNRgrid deployed MRs, per-MR capacity with intracluster APtraffic is NAB

NR, ifBOW.

Proof. When the access link data rate is BOW, thecapacity is constrained by the access links. The through-put is no more than the total throughput of all APs andthe IGW. While each AP-MR fully utilizes access linkbandwidth, the per-MR capacity is

RpBNA W

NR: 19

While the capacity is constrained by the access link,the backbone link bandwidth is sufficient for trafficbalance among the MRs having AP access. Without lossof generality, we assume that the traffic are toward theAPs. For an AP-MR cell, the feasible solution can redirectthe traffic in the cell with AP access to the AP in the samecell. The rest of the cells without AP access, routes thetraffic to the cells with AP access by considering trafficbalance among different AP-MR cells.

In the cell having AP access, the AP access link sharesthe bandwidth with MCs. The aggregated traffic betweenthe MR and the MCs takes Rp bandwidth, and theredirection of this traffic to the AP costs additional Rp

bandwidth. The residual bandwidth is available for otherMRs. As we can evenly associate non-AP-MRs to eachAP-MR or the IGW, the backbone bandwidth is not aconstraint. The number of associated non-AP-MRs forevery AP-MR is no more than NRNA1

NA1 1. The asymp-totic capacity is

RpNR

NA 1B 2Rp ;

Rp BNA1NR 2NA 2 :

20

From (19) and (20), the asymptotic capacity of

BNA

NR holds for B

O

W

. tuLemma 5. Given a connected backbone network having NR

grid deployed MRs, per-MR capacity with intracluster APtraffic is NAW

NR, if BomegaW and NAO


p .

Proof. While B!W, the intracluster AP throughput isconstrained by backbone links rather than access links.The aggregated traffic at the AP-MR cell would not beforwarded into backbone network but redirected to theAP by the MR via the access link. The upper bound ofthe capacity is

Rp

NA 1WNR NA 1

:

21

The interference at neighboring AP-MRs could de-

grade the throughout. With grid topology, each MR will

have four neighboring MRs that could interfere with it. If

AP-MR has four neighboring MRs that all have AP

accesses, the equal bandwidth shared among five MRs

will be less than W5

. It degrades throughput if the traffic is

relayed from peripheral AP-MRs to the middle AP-MR.

So, we do not allocate the traffic from the non-AP-MRs to

the AP-MRs that are surrounded by AP-MRs. The traffic

is only allocated and balanced among the AP-MRs

having non-AP-MRs available as neighbors. The prob-ability that a backbone link is adjacent to two AP-MRs,

is approximated byN2

A

N2R

. According to the Chernoff upper

tail bound, the number of links adjacent to two AP-MRs,

denoted bynA, is bounded by

P nA 1 c2 N2A

NR

e

c22

N2A

2c2NR ; 22

where c2 is a positive constant.

If NAOffiffiffiffiffiffiffi

NRp , nA is bounded by1 c2 N

2A

NRwith

high probability (w.h.p.). The number of AP-MRs

surrounded with AP-MRs is less than nA. We use thesame feasible traffic distribution and balance them as

illustrated by Lemma 4 for the intracluster AP traffic. The

feasible throughput per MR is

RpWc1

NA 1c2N

2A

NR

NR NA 1 : 23

To summarize, per-MR capacity with intracluster AP

traffic isRpWNANR , ifB !W andNA Offiffiffiffiffiffiffi

NRp .tu

Lemma 6.Given a connected backbone network having NR griddeployed MRs, per-MR capacity with intracluster AP traffic

is NAWNR , ifB!W and NA! ffiffiffiffiffiffiffiNRp .Proof.The upper bound for per-MR capacity is

Rp NA 1WNR NA 1 : 24

The probability that a backbone link is adjacent to anon-AP-MR and an AP-MR is 2NA

NR1 NA

NR.

According to the Chernoff bound, the number of suchlinks in the network has a lower bound of

P nM 1 c32 NA N2A

NR ec2

3NA

N2A

NR; 25

where c3 is a positive constant less than 1.Since NA!


p , nM is no less than c4NA N2A

NR

w.h.p., where c421 c3. Several links may have the



8/15

same adjacent AP-MR, but the maximum number oflinks being adjacent to the same AP-MR is no more thanfour. So, the number of AP-MRs has non-AP-MR in onehop is more than c4

4NA N

2A

NR w.h.p. In the worst case,

we can schedule the traffic to these AP-MRs havingneighboring non-AP-MR. An AP-MR can provide at leastthroughput of W

c1bps. With the feasible scheduling, the

per-MR capacity is

Rpc4

NA N2A

NR

W

4c1NR NA 1 : 26

In summary, Rp NA WNR , if B!W and NA! ffiffiffiffiffiffiffiNRp . tuProposition 2 is obtained from Lemmas 4, 5, and 6.

Proposition 2.Given a connected backbone network consisting of

NRgrid deployed MRs, per-MR capacity with intracluster AP

traffic is

Rp

NAB

NR

; BOW;

NAW

NR

; B!W:

8>>>: 27

3.4 Intercluster Traffic through IGW

Assuming that intercluster traffic only transmits in thebackbone toward or from the IGW, the per-MR capacity isbounded by the bandwidth of the IGW and given by thefollowing proposition:

Proposition 3. Per-MR capacity of intercluster traffic without

AP connections is

Rg W

NR

: 28

Proof. When the intertraffic goes through the IGW, thecapacity is constrained by the bandwidth at the IGW andthe wired connection of the IGW is assumed to haveunconstrained bandwidth. The bandwidth can be fairlyshared by all the MRs in the cluster and the per-MRthroughput is

Rg W

NR 1 : 29IGW shares the bandwidth with the neighboring MRs.

With the channel assignment, the maximum number of

required channels is no more than c1. The feasible IGW

throughput is no less than 4Wc1

bps. The assigned traffic at

each MR is Rg 4Wc1NR1 . MR can have throughput thatis more than 4W

c1. Considering the relay and aggregated

traffic, the traffic at MR u follows:

Rg Wiu Wou 4W

c1;

Rg Wiu Wou;8>>>>: 33

Proof.While the capacity is constrained by the access linksof the AP-MRs, we have

ReNR NA 1 NAB Re;Re BNANR 1 : 34

Otherwise, it is constrained by the backbone. The feasiblerouting and scheduling can use a similar methodindicated in the proof of intracluster traffic goingthrough APs to reach the capacity bound. With such atraffic balance, the traffic from or to backbone is given byRe

NRNA1NA

. A feasible scheduling is

ReNR NA1

NA 2Re B;

Re BNA

NR

NA

1:

35

When the capacity is constrained by the backbone,there are two situations: NAO


p and NA! ffiffiffiffiffiffiffiNRp . When NAO ffiffiffiffiffiffiffiNRp , the APs are far apart



9/15

and do not interfere with one anotherw.h.p., as given byLemma 5. Then, the asymptotic capacity isRe !NAWNR .

When NA!ffiffiffiffiffiffiffi

NRp , some MRs having APs are

surrounded by other AP-MRs and cannot effectivelyprovide the traffic shortcut. However, the rest of AP-MRscan be accessed by non-AP-MRs and the capacity canreachNAW

NRas proved in Lemma 6. tu

4 PER-MC CAPACITY

The per-MC capacity is not only constrained by the

backbone per-MR capacity but also the access link of each

cell. The asymptotic capacity can be identified by a

combination of traffic under various cases.The main results of per-MC capacity are summarized

here. Per-MC capacity with MR ad hoc routing is given by

Cr

NRB

NC

; NR O NC

logNC

^ Bo W


p

;

ffiffiffiffiffiffiffiNRp W

NC

; NR O NC

logNC ^ B WffiffiffiffiffiffiffiNRp ;

B

logNC

; NR ! NC

logNC

^ Bo Wffiffiffiffiffiffiffi

NRp

;

WffiffiffiffiffiffiffiNR

p logNC

; NR ! NC

logNC

^ B Wffiffiffiffiffiffiffi

NRp

:

8>>>>>>>>>>>>>>>>>>>>>>>:

36Per-MC capacity with intracluster AP traffic is obtained as

Cp

NAB

NC

; BOW;

NAW

NC

; NAO NC

logNC ^ B!W;

W

logNC

; NA! NC

logNC

^ B!W:

8

>>>>>>>>>>>>>>>:37

The intracluster traffic is the combination of ad hoc MR

traffic and intracluster AP traffic. Considering reasonable

access link bandwidth B WffiffiffiffiffiNR

p ^ BOW, per-MCcapacity with intracluster traffic can be expressed by

CI


p W

NC

; NRO NC

logNC

^NAO WffiffiffiffiffiNRpB ;

NAB

NC

; NRO NC

logNC

^NA! Wffiffiffiffiffi

NRpB

;


p logNC

; NR! NC

logNC

^NAO W NCBffiffiffiffiffiNRp logNC

;

NAB

NC

; NR! NC

logNC

^NA! W N

CBffiffiffiffiffi

NRp logNC

:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

38

Per-MC capacity with IGW traffic isCg WNC. Per-MCcapacity with intercluster AP traffic is given by

Ce

NAB

NC

; BOW;

NAW

NC

; NAO NC

logNC

^ B!W;

W

logNC

; NA! NC

logNC

^ B!W:

8>>>>>>>>>>>:

39

Per-MC capacity with intercluster AP traffic, if BOW, is expressed by

CE NAB

NC

: 40

Besides dominating factor of the number ofNR and/orNA, the ratio of B to W also has a strong impact on theselection of capacity region. As there are two spectrums in aWMN, the network bottleneck could be either the backbonelink bandwidth or the access link bandwidth. For appro-priate capacity region, the ratio ofBtoWshall be

B

W

1

ffiffiffiffiffiffiffiNRp ^

O

1

:

41

Per-MC capacity within the region of (41) is expressed by

C


p W

NC

; NRO NC

logNC

^NAO Wffiffiffiffiffi

NRpB

;

NAB

NC

; NRO NC

logNC

^NA! Wffiffiffiffiffi

NRpB

;

WffiffiffiffiffiffiffiNRp logNC ; NR!

NC

logNC

^NAO WNCBffiffiffiffiffiNRp logNC

;

NAB

NC

; NR! NClogNC

^NA! WNCBffiffiffiffiffiNRp logNC

:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

42

The access link bandwidthBneeds to be higher or equal

order of WffiffiffiffiffiNR

p ; otherwise, the access link is the bottleneck andstrongly constraints the capacity. While B is the higher

order of W, W becomes the capacity constraint. When

B

Wffiffiffiffiffi

NRp ^B

O

W

, the question about which link

serves as the capacity constraint will also depend on the

number of random APs. Equation (42) shows the capacity

region for BW

between 1ffiffiffiffiffiNR

p and O1. While NRO NC

logNC, the capacity can be enlarged with NA!W

ffiffiffiffiffiNR

pB

and capacity is constrained by access link bandwidth B.

While NR ! NClogNC, the capacity can be significantlyincreased with NA! WNCBffiffiffiffiffiNRp logNC.4.1 Per-MC Capacity with MR Ad Hoc Routing

Lemmas 7, 8, and 9 give the bounds for the number of MCsin a cell with different orders of the number of MRs, which

lead to different per-MC asymptotic capacity in laterpropositions. Lemma 10 is the critical transmission distance

for MCs in a unit square. It serves as the bound for thedistance between any two MCs in the region.



10/15


11/15

within a circle with radiusan w.h.p. For the intraclustertraffic, it is generated by randomly picking up two points

in the unit square and the nodes closest to the points are

selected as the source and the destination, respectively.

So, the MCs in the cell will be selected as the destination

nodes while random destination points fall in the

combined area of the cell and the surround region with

an width. The size of such area can be computed by

Pd 1ffiffiffiffiffiN

pR

2an 2

1NR

4a2n 4anffiffiffiffiffiffiffiNR

p :55

Downlink traffic satisfies Cr PdNCniCminfR; Bg.We have

Cr minfR; BgPdNC niC

minfR; Bg 1NR

4a2n 4anffiffiffiffiffiNR

p

NC1 o1

minfR; BgNCNR

4c20log NC

4c0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

NClog NCNR

q 1 o1 :56

IfNRO NClogNC, we obtain

Cr c9NR

NCminfR; Bg: 57

Let nD denote the number of MCs selecting the MCsin a cell as the destination. With the Chernoff upperbound, we have

P nDc10NCNR

e

c10

NR Pd12

1 c10NR Pd

NCPd

; 58

where c10 > 1.IfNRO NClogNC, the RHS of the equation approaches

zero, sonDis upper bounded byc10NC

NRw.h.p. With feasible

TDMA scheduling, we have

Cr minfR; Bg

nD

NRminfR; Bg

c10NC

: 59

Thus, Cr

NR

NC

min

fR; B

gholds,ifNR

O

NC

logNC.tuProposition 8. Downlink traffic of MCs at a cell is Cr

minfR;BglogNC

, ifNR! NClogNC.Proof. According to Lemma 10, the maximum per-MC

capacity is upper bounded by

CrminfR; Bg

c7log NC; 60

ifNR! NClogNC.Also, from the Chernoff upper bound, the number of

MCs having traffic to a cell is upper bounded by

PnDc11logNC e c11NClogNCPd1

2NCPd

1 c11NClogNCPd ; 61

where c11 > 1.

The RHS of the equation approaches zero if

NR! NClogNC. So, nD is upper bounded by c11log NCw.h.p. With the feasible TDMA scheduling, we have

Cr minfR; Bg

nD

minfR; Bg

c11logNC

: 62

Thus,Cr

minfR;BglogNC

holds, ifNR

!

NClogNC

.

tuSubstituting ad hoc per-MR capacity by the expression

obtained from previous section, the per-MC intraclustertraffic with ad hoc backbone routing is

Cr

NRB

NC

; NRO NC

logNC

^Bo WffiffiffiffiffiffiffiNR

p

;


p W

NC

; NRO NC

logNC

^B Wffiffiffiffiffiffiffi

NRp ;

B

logNC

; NR! NC

logNC

^Bo WffiffiffiffiffiffiffiNR

p

;


p logNC

; NR! NC

logNC

^B WffiffiffiffiffiffiffiNR

p

:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

63

4.2 Per-MC Capacity with Intracluster AP Traffic

While utilizing a random AP connection, the ratio of thebackbone link bandwidth to the access link bandwidth isvery critical. The traffic via the APs employs the access link.While it is not sufficient, the benefits from using the randomconnections will be limited. An increase in the number ofAP-MRs can help increase the traffic through the APs. Theefficiency of implementing a large number of APs dependsvery much on the strategies of traffic load balancing. Ourtraffic balance is based on MCs rather than MR cells. Thetraffic from the same MR cell may go to different AP-MRsby considering the traffic load on the AP-MRs.

Proposition 9 is for BOWcase, where the backbonelink bandwidth is assumed sufficient for the traffic balanceamong different AP-MRs. The per-MC capacity with B!W is considered in Propositions 10 and 11 for NAO NC

logNCandNA! NClogNC, respectively.

Proposition 9.IfBOW, per-MC capacity with intraclustertraffic having random AP access is

Cp NAB

NC

: 64

Proof. Because the traffic can be evenly distributed in thebackbone for intracluster AP traffic, we analyze the

traffic from MCs to APs without any loss of generality.The traffic of an AP from the MR includes the aggregatedtraffic from MCs in the cell and the traffic from othernon-AP-MRs. Non-AP-MRs intracluster AP traffic can



12/15


13/15

From Lemma 10, there exists an MC that is selected byc7logNCMCsw.h.p. So, we have the upper bound of thecapacity as

Cp B

c7logNC: 76

When the upper bound of the number of MCs in a cellisc6logNC, it does not collect the traffic from other non-AP-MR cells because its traffic is above the average. So,the feasible uplink per-MC capacity is

Cp B

2c6logNC

: 77

For the downlink traffic, the analysis is similar to theproof of Proposition 8 and we also use the upper boundof nD. The difference is that the traffic is from the APthen to the MR and does not pass through the backbonewhile nD approaches its upper bound. We obtain

Cp

B

2nD

B

2c11logNC : 78

Thus, the proposition is satisfied. tu

In general, access link bandwidth B WffiffiffiffiffiNR

p . So, theaccess link would not be a bottleneck for the per-MC capacity

in ad hoc routing. In this region, the per-MC capacity is

Cr


p W

NC

; NRO NC

logNC

^B W


p

;

WffiffiffiffiffiffiffiNRp logNC

; NR!

NC

logNC

^B WffiffiffiffiffiffiffiNR

p

:

8>>>>>>>>>>>>>>>>>>>>>:

79

Since access to APs are random, access link bandwidth isdesigned according to the analysis and requirements of aWMN, which is BOW. In this region, the per-MCcapacity with AP routing is

Cp NA

NCB

; BOW: 80

4.3 Per-MC Capacity with Intercluster Traffic

Following a similar process of MR intercluster capacity, theMC intercluster capacity can be derived as follows:

Proposition 12. Per-MC capacity with intercluster trafficthrough the IGW is Cg WNC.

Proof.The proof is similar to Proposition 4. tuProposition 13.Per-MC capacity with intercluster AP traffic is

expressed by

Ce

NAB

NC

; BOW;

NAW

NC ; NAO

NC

logNC ^ B!W;

W

logNC

; NA! NC

logNC

^ B!W:

8>>>>>>>>>>>>>:

81

Proof.Due to the traffic balance and the AP routing scheme,per-MC capacity computed for intracluster AP traffic isapplicable to intercluster AP traffic. The difference is thatintercluster AP traffic needs to consider both the capacityconstraints as the source and the destination are in thesame cluster. So, we use (70) under case B!W toderiveCe WlogNC. The rest procedures are similar toProposition 11.

tu5 DISCUSSION AND IMPLICATIONS

MRs can be deployed on electric poles or on the top of houseroofs and can cover hole and/or network area extension.The power supply for MR operations can come from theelectric poles or the solar panels. With the wireless relay androuting function, MRs do not need wired cables except forthe IGWs. The feasibility of deployment enables WMNs tocover any such desired area to take care of decay in thesignal strength and environmental complexity in an urbanarea. Throughput of MCs can be improved by accessing

deployed MRs. MCs can have much better Signal toInterference Noise Ratio (SINR) and higher data transmis-sion rate by accessing nearby MRs. With preplanned andoptimized deployment, connections among MRs shouldhave a required data transmission rate. By controlling thecoverage of MRs, the maximum transmission between MCand its serving MR is also determined, which affects thequality of the signals and the service. Overall, the systemperformance can be improved with specific benefitsdiscussed above. It is also one of the reasons that IEEE802.16j [18], IEEE 802.16m [19], and LTE Advanced [20] haveadopted relay stations in their systems to fill the cellcoverage holes and/or enhance the cell capacity.

Grid topology provides a better coverage and anenhanced capacity than that of a random topology. Griddeployment discussed in [21] argues for a better coveragethan a hexagonal deployment. While MRs are deployed ingrids, it also naturally matches grid street topology likeManhattan model. Backbone link between two neighboringMRs can be within the line of sight and have better channelcondition. On the other hand, grid deployed MRs canprovide better access for MCs along the streets.

In our analytical work, backbone links employ differentfrequency bands from that of the access links. Backbonelinks among MRs may use licensed band so as to reduce anyunexpected interference and guarantee the backbone

performance. The stability and quality of backbone linksare essential for the quality of service provided to the MCs.Most of the traffic is delivered via the backbone links. Apoor backbone link may drop traffic flows and lead to chaosin the network. So, MRs are mostly deployed by thenetwork operators. They use licensed band that is protectedand seldom interfered by unlicensed band used by wirelesshome phone, various monitoring system, WLANs, and soon. Some present public places may have tens of differentsystems using the same unlicensed band. If unlicensed bandis used for a backbone link, signals will largely interferewith the transmission of a WMN backbone. In contrast, tofacilitate the access of MCs, it is very likely to use unlicensed

band and protocols for access link of MR cells. To becompatible with many current mobile devices, contentionbased protocols like IEEE802.11 are most likely to be usedfor access links between MRs and MCs. Roaming MCs can



14/15

recognize MR cells and connect to them for access. WhenMC moves from a cell to a neighboring cell, the backboneperforms the handover to maintain service continuity forthe MC.

WMN is a cost efficient way to provide network accessservice for a wide region. However, the capacity constraintdue to multiple hop transmission is still the drawback toovercome before having a wider acceptance. As discussedearly, the per-MR capacity is

WffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNRlog NR

p

:

By optimizing the placement, it approaches WffiffiffiffiffiNR

p . Utilizingrandom connection to APs can improve the capacitysignificantly, especially when the number of AP-MRs isincreased. With a large number of MRs and WLAN APs thathave been deployed, the average number of such APconnections is large. Per-MR capacity can be increased upto NABW

NR. Because random connection is utilized only for a

single hop between MR and AP, it is still highly reliable ascompared to a pure multihop random network.As an Internet gateway, the backbone link bandwidth W

of the IGW is constrained by the maximum throughput ofthe intercluster traffic if without random AP access. Itserves as the lower bound of intercluster traffic perfor-mance if there is no constraint on the access link bandwidth.By adjusting the number of IGWs deployed in a region, itcontrols the coverage area of every cluster led by an IGW.While the density of MCs is given, smaller cluster coveragearea means smaller number of MCs in a cluster and moreintercluster traffic per MC can be supported. An extremecase of increasing the number of IGWs is to form a multicell

single hop network architecture like GSM base stations.This is not as cost-efficient because the cost to deploy such anetwork is NIGW.

Given a cluster with fixed coverage, increasing the densityof MRs could improve the per-MC capacity with intraclustertraffic. Assume that the access link bandwidth is not thebottleneck, i.e., B Wffiffiffiffiffi

NRp . The increase in the MR density

have two different effects: when the number of MRs in aunit square is less than a threshold ofO NC

logNC, the per-MC

capacity increases withffiffiffiffiffiffiffi

NRp

. When the number of MRsreaches or runs over the threshold, the per-MC capacity withintracluster traffic starts to degrade. Thus, it is important to

consider the upper bound and choose an appropriatenumber of MRs for an optimal performance.Random access to APs has positive impact on both

intercluster traffic and intracluster traffic. With access toAPs, intracluster traffic has the routing shortcut through thewired connection between APs. Intracluster traffic path likeMC-(-MR)k-MC can be changed like MC-(-MR)i-AP-AP-(-MR)j-MC, where i, j, and k denote the number of inter-mediator MRs and i j < k. In that case, not only thecapacity can be improved but also the delay can bedecreased. The wired connection is assumed to have enoughthroughput and the transmission delay between two APs isless thank

i

j

hops. Per-MC capacity with intercluster

traffic is also improved with AP access. Intercluster trafficwould not be constrained by the IGW, as the traffic can gothrough the APs and the wired connection of APs to otherclusters or the Internet.

It is noticeable that the wireless access between MRs andAPs use the same type of air interfaces as that between MRsand MCs. So, it is easy for MRs to utilize the APs that arewidely deployed, without drastic changes in the networkfunctions. MR can take the APs as special MCs. Thedifference is that the MR needs to be aware of wiredconnection of APs to the Internet and redirect the traffic tothe APs or in the reverse direction. The reasons that MCs in

an AP-MR cell connect to the MRs rather than to the APs inthe cell directly, differ on the following aspects. APs usuallyassume the access clients are static while MRs assume theMCs having mobility and provide better channel estimatefor optimizing the link performance. MCs can have betterresponse if connected to the MR. MRs can perform accesscontrol for QoS and security purposes. It is unconventionalto have such functions for the random access APs. Inaddition, the handover for MCs can be supported by MRs.With the information exchange in the backbone, MCs areable to handover from the serving MR to a neighboring MR.These functions are naturally supported by the MRs. APs

are only able to serve as tunnels for the traffic.The number of AP-MRs has a different level of impact on

the capacity. For intracluster traffic, the capacity of

intracluster traffic increases withffiffiffiffiffiffiffi

NRp

if NR O NClogNC.With the random access to APs, the capacity is improved by

the number of AP-MRs. IfNAOWBffiffiffiffiffiffiffi

NRp , the capacity is

still predominated by the traffic with ad hoc routing in the

backbone. However, ifNA!WBffiffiffiffiffiffiffi

NRp , the capacity is then

dominated by random access to the APs and the asymptotic

capacity becomes NABNR

. On the other hand, if there is alarge number of MRs in the cluster that NR! NClogNC, NAwould have similar effect. IfNA ! W NCBffiffiffiffiffiNRp logNC, the asymp-totic capacity is

NA

BNC . For the intercluster traffic, the

increase of MRs accessing to APs almost linearly increase

the asymptotic capacity while access link is not the

bottleneck. Under these situations, increasing cell access

link bandwidth will almost linearly increase the per-MC

capacity. This is a significant enhancement to the capacity.In this three-tier network, backbone link bandwidth W,

and access link bandwidthB, not only their absolute value

but also their ratio, determine the bottleneck of the network.

For the intracluster traffic, the capacity will be bounded by

the access link ifBo W

ffiffiffiffiffiNRp . Then, the per-MC capacity is

determined by the number of MCs in a cell, which leads to

the asymptotic capacity NRBNC

. Under this situation, theaccess to APs does not have much effect on the capacity.

When B WffiffiffiffiffiNR

p and NAOWffiffiffiffiffi

NRpB

, access link is nolonger the bottleneck. When NA!W

ffiffiffiffiffiNR

pB

, the role ischanged again and the increase for the bandwidth of access

link will increase the capacity. Thus, the access link

bandwidth should be large enough so that it would not be

the bottleneck of the network. According to the derivation in

previous sections, B WffiffiffiffiffiNR

p ^ B OW. IfB o WffiffiffiffiffiNR

p , thenetwork capacity will be seriously constrained by

the access link bandwidth. If B !W, access linkbandwidth will be wasted if there are no enough randomAPs connections. While taking B Wffiffiffiffiffi

NRp ^ BOW,

the values ofNR, NA, W,and Bmutually affect each other for

the asymptotic per-MC capacity.



15/15

6 CONCLUSION

In this paper, we have derived asymptotic capacity of a

hybrid WMN architecture with random connections to APs.

With this, the access link bandwidth greatly affects the

capacity, which is different from a conventional WMN. Tosome extent, the ratio of access link bandwidth to backbone

link bandwidth is critical. It dominates the capacity bottle-

neck and magnifies the influence of AP-MRs. Theoreticalresults show that thecapacity enhancement by accessing APswithin the coverage of MRs is significant with an increase in

the number of AP-MRs and the bandwidth ratio. The

improvement is at the very low cost by utilizing currently

available networks and it is also possible for those networksto take advantage of a WMN. It should be noted that the

access to the APs are random, it may have negative impact on

WLANs performance when the WLANs traffics are heavy.In the future work, we plan to consider the traffic between

MRs and WLANs so as to have a control mechanism that

takes the traffic load at the APs into account.

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[11] M. Kodialam and T. Nandagopal, Characterising the CapacityRegion in Multi-Radio Multi-Channel Wireless Mesh Networks,Proc. ACM MobiCom,2005.

[12] D.P. Agrawal and Q. Zeng, Introduction to Wireless and MobileSystems. Thomson Brooks/Cole, 2003.

[13] P. Gupta and P. Kumar, Internets in the Sky: The Capacity ofThree Dimensional Wireless Networks,Comm. in Information andSystems, vol. 1, pp. 33-50, 2001.

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Weihuang Fu received the BS degree incommunications engineering and the MS degreein communication and information systems fromthe Zhejiang University of Technology, China, in2002 and 2005, respectively, and the PhDdegree in computer science and engineeringfrom the University of Cincinnati, Ohio, in 2010.He joined Cisco Systems, Inc., in 2010. He hasbeen a technical program committee member ofmany international conferences and a technical

reviewer of numerous international journals and conference proceedings.His research interests include network modeling and analysis, MAC/routing protocol for wireless multihop networks, and mobility manage-ment. He is a member of the IEEE and the IEEE Computer Society.

Dharma P. Agrawal is the Ohio Board ofRegents distinguished professor in the Schoolof Computing Sciences and Informatics and thefounding director for the Center for Distributedand Mobile Computing at the University ofCincinnati, Ohio. He was a visiting professor ofelectrical and computer engineering at theCarnegie Mellon University on sabbatical leaveduring the autumn 2006 and winter 2007quarters. He was a faculty member at North

Carolina State University, Raleigh from 1982-1998 and at Wayne StateUniversity, Detroit from 1977-1982. His current research interestsinclude energy-efficient routing and information retrieval in sensor andmesh networks, QoS in integrated wireless networks, use of smartmultibeam directional antennas for enhanced QoS, various aspects ofsensor networks including environmental monitoring, and secured

communication in ad hoc and sensor networks. His coauthored textbook,Introduction to Wireless and Mobile Systems, third edition published byCengage Corp., has been adopted throughout the world and revolutio-nized the way the course is taught. His second coauthored textbook,Ad Hoc and Sensor Networks, second edition, was published by WorldScientific. He has served as an editor of the IEEE Computer, the IEEETransactions on Computers, and theInternational Journal of High SpeedComputing. He is an editor for the Journal of Parallel and DistributedSystems, International Journal on Distributed Sensor Networks, Inter-national Journal of Ad Hoc and Ubiquitous Computing (IJAHUC),International Journal of Ad Hoc and Sensor Wireless Networks, andJournal of Information Assurance and Security. He has been theprogram chair and general chair for numerous international conferencesand meetings. He has received numerous certificates and awards fromthe IEEE Computer Society. He was awarded a Third MillenniumMedal by the IEEE for his outstanding contributions. He has also

delivered keynote speeches for five international conferences. He hasfive patents and 23 patent disclosures in the area of wireless networking.He served as a fulbright senior specialist for five years and was thefounding editor-in-chief of the Central European Journal of ComputerScience, Versita. He has graduated 62 PhD and 51 MS students andwas also named as an ISI Highly Cited Researcher in computer science.He won the 2008 Harry Goode Memorial award from the IEEE ComputerSociety and the 2011 Award for Excellence in Mentoring of DoctoralStudents, University of Cincinnati. He is a fellow of the IEEE, IEEEComputer Society, ACM, AAAS, and WIF.


Documents

Capacity of Hybrid Wireless Mesh