1106 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 17, NO. … · 1106 IEEE/ACM TRANSACTIONS ON...

1106 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 17, NO. 4, AUGUST 2009

Asymptotic Connectivity in Wireless Ad HocNetworks Using Directional Antennas

Pan Li, Student Member, IEEE, Chi Zhang, Student Member, IEEE, and Yuguang Fang, Fellow, IEEE

Abstract—Connectivity is a crucial issue in wireless ad hocnetworks (WANETs). Gupta and Kumar have shown that inWANETs using omnidirectional antennas, the critical transmis-sion range to achieve asymptotic connectivity is � �� if

nodes are uniformly and independently distributed in a disk ofunit area. In this paper, we investigate the connectivity problemwhen directional antennas are used. We first assume that eachnode in the network randomly beamforms in one beam direction.We find that there also exists a critical transmission range for aWANET to achieve asymptotic connectivity, which correspondsto a critical transmission power (CTP). Since CTP is dependenton the directional antenna pattern, the number of beams, andthe propagation environment, we then formulate a non-linearprogramming problem to minimize the CTP. We show that whendirectional antennas use the optimal antenna pattern, the CTPin a WANET using directional antennas at both transmitter andreceiver is smaller than that when either transmitter or receiveruses directional antenna and is further smaller than that whenonly omnidirectional antennas are used. Moreover, we revisit theconnectivity problem assuming that two neighboring nodes usingdirectional antennas can be guaranteed to beamform to eachother to carry out the transmission. A smaller critical transmis-sion range than that in the previous case is found, which impliessmaller CTP.

Index Terms—Wireless ad hoc networks, directional antenna,asymptotic connectivity, critical transmission range, critical trans-mission power.

I. INTRODUCTION

F OR a long time, people have dreamed of breaking throughthe limitation of physical distance and communicating

with each other tetherlessly anytime at any place. Now, thisdream is being realized step by step with the rapid developmentand deployment of wireless networks, such as wireless localarea networks (WLANs), wireless ad hoc networks (WANETs)including mobile ad hoc networks (MANETs) and wirelesssensor networks (WSNs) [1], and wireless mesh networks

Manuscript received April 27, 2007; revised March 02, 2008; approved byIEEE/ACM TRANSACTIONS ON NETWORKING Editor N. Shroff. First publishedDecember 31, 2008; current version published August 19, 2009. This workwas supported in part by the National Science Foundation under Grants CNS-0721744 and DBI-0529012. The work of Y. Fang was also supported in part bythe 111 Project under B08038 with Xidian University, China. A preliminary ver-sion of this paper was presented at the IEEE Int. Conf. Distributed ComputingSystems (ICDCS), Toronto, Ontario, Canada, June 25–29, 2007.

P. Li and C. Zhang are with the Department of Electrical and ComputerEngineering, University of Florida, Gainesville, FL 32611 USA (e-mail:[email protected]; [email protected]).

Y. Fang is with the Department of Electrical and Computer Engineering, Uni-versity of Florida, Gainesville, FL 32611 USA, and also with the National KeyLaboratory of Integrated Services Networks, Xidian University, Xi’an, China(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNET.2008.2006224

(WMNs) [2]. Although omnidirectional antennas are com-monly used in these networks, directional antennas have gainedtremendous attention due to the increased transmission range,the improved spatial reuse, and the decreased interference.

Up to now, many research works on wireless networksusing directional antennas focus on the design of MAC pro-tocols [3]–[13]. There are relatively few works on networkconnectivity, which is indeed one important problem. Guptaand Kumar [14] study the connectivity problem for wirelessnetworks using omnidirectional antennas and show that thereis a critical transmission range for a networkto achieve asymptotic connectivity when there are nodesuniformly and independently distributed in a disk of unit area.This problem has not been studied when directional antennasare used except in our preliminary work [15].

In this paper, we address the connectivity problem inWANETs using directional antennas. Other than using thesimple sector model, we introduce a more realistic directionalantenna model, which consists of one main lobe with main lobeantenna gain , as well as side lobes withthe same side lobe antenna gain . We show that the side lobeantenna gain does have a significant impact on the networkconnectivity, which cannot be simply neglected. In our model,directional antennas can work either in the directional mode, orin the omnidirectional mode. So, according to the usage of di-rectional antennas, we can classify WANETs using directionalantennas into four categories: DTDR (Directional Transmissionand Directional Reception) networks, DTOR (DirectionalTransmission and Omnidirectional Reception) networks, OTDR(Omnidirectional Transmission and Directional Reception)networks, and OTOR (Omnidirectional Transmission and Om-nidirectional Reception) networks. Note that OTOR networksare exactly the networks Gupta and Kumar study in [14]. Wealso note that for directional antennas, with fixed transmissionpower, the directional transmission range is directly dependenton the omnidirectional transmission range given the directionalantenna pattern , the number of beams , and thepath loss exponent .

Assuming every node in a WANET randomly beamforms inone beam direction, we derive the necessary and sufficient con-ditions for the WANET to achieve asymptotic connectivity. Wefind that there also exists a critical omnidirectional transmis-sion range. Thus, we can compare the power consumption whendirectional antennas are used with that when omnidirectionalantennas are used, i.e., in OTOR (Omnidirectional Transmis-sion and Omnidirectional Reception) networks, by simply com-paring their critical omnidirectional transmission ranges. Forsimplicity, we call critical transmission range (CTR) instead.

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LI et al.: ASYMPTOTIC CONNECTIVITY IN WIRELESS AD HOC NETWORKS USING DIRECTIONAL ANTENNAS 1107

Specifically, we show that compared to the critical transmis-sion range in OTOR networks, the critical transmission

range is in DTDR networks, in DTOR net-

works, and in OTDR networks, respectively, where, and

. When , the critical transmission rangein DTDR networks is smaller than that in OTOR networks.So, the corresponding transmission power for achieving asymp-totic connectivity, which we call critical transmission power(CTP), in DTDR networks is also smaller than that in OTORnetworks. DTOR (or OTDR) networks have the same situationif . Then, a question arises: how low can the crit-ical transmission power level go?

In order to minimize the CTP in DTDR, DTOR, and OTDRnetworks, respectively, we formulate a non-linear program-ming problem to find the optimal directional antenna pattern

, given the number of beams and the path lossexponent . We also investigate the impacts of and on thenetwork connectivity. We show that in the outdoor environ-ments , with the same antenna pattern ,and the same number of beams , when the CTPin DTOR or OTDR networks is smaller than that in OTORnetworks, the CTP in DTDR networks is further smaller thanthat in DTOR or OTDR networks. However, when the CTPin DTOR or OTDR networks is larger than that in OTORnetworks, the CTP in DTDR networks is further larger thanthat in DTOR or OTDR networks. Moreover, with the sameand , the minimum CTP, which is achieved when directionalantennas use the optimal antenna pattern, in DTDR networks isalways smaller than that in DTOR and OTDR networks, whichis further smaller than that in OTOR networks when .When , the minimum CTP in DTDR networks is thesame as that in DTOR, OTDR, and OTOR networks. Besides,as increases, or decreases, the minimum CTP decreases,which means we can save more power.

Furthermore, we revisit the connectivity problem in the morerelaxed sense that there exists a schedule according to whichtwo neighboring nodes can be guaranteed to beamform to eachother to carry out the transmission. A smaller critical transmis-sion range than that in the previous case is found, which impliessmaller CTP.

The rest of this paper is organized as follows. We reviewthe related work in the next section. In Section III, we intro-duce our directional antenna model and the power propagationmodel. Section IV presents the necessary and sufficient condi-tions for WANETs using directional antennas to achieve asymp-totic connectivity. In Section V, we illustrate how to find theoptimal directional antenna pattern in order to attain the asymp-totic connectivity with the smallest transmission power. We alsodiscussed the impacts of the number of beams and the path lossexponent on the network connectivity. Section VI gives some in-sights on our results. An extended case is studied in Section VII.We finally conclude this paper in Section VIII.

II. RELATED WORK

How to achieve connectivity is a very important problem inWANETs. Gupta and Kumar [14] show that for a network of

uniformly and independently distributed nodes to achieveasymptotic connectivity, there is a CTR required for connec-tivity, which is . While the work in [14] assumesthe nodes in the network are static, Madsen et al. [16] studythe connectivity probability in mobile ad hoc networks, wherethe position of the nodes and the link quality change overtime. Some stationary mobility models are considered there,including random direction model, random waypoint model,attractor model, virtual world model, and mobility models withobstacles. Dousse et al. [17] study the impact of interferenceson connectivity in ad hoc networks and show that there is acritical value of above which the network is made of discon-nected clusters of nodes, where is a coefficient that weighsthe effect of interferences. Recently, Yi et al. investigate theasymptotic connectivity problem for ad hoc networks withBernoulli nodes in [18] and [19].

Other than asymptotic connectivity discussed in the worksabove, Balister et al. [20] derive for the first time reliable den-sity estimates for thin strip networks with limited number ofnodes to achieve connectivity. They also demonstrate the ac-curacy of the estimates through simulations. [20] represents avery useful work for practical deployment and bridges the gapbetween theory and practice. Since we are envisioning a wire-less network with a very large number of nodes in the future,we will focus on asymptotic connectivity in this paper as that in[14].

Notice that all the works above assume the use of omnidi-rectional antennas in the networks. There are also some worksrelated to connectivity for networks using directional antenna.Kranakis et al. [21] study the -connectivity problem in wire-less sensor networks and only derive sufficient conditions on thebeamwidth of directional antennas so that the energy consump-tion required to maintain k-connectivity is lower when using di-rectional antennas than when using omnidirectional antennas.Diaz et al. [22] investigate the value of the chromatic number

, the directed clique number , and the undirectedclique number for random scaled sector graphs.1 Theyshow that when , where is the beam width of directionalantennas, and are with high proba-bility (w.h.p.), while is w.h.p., as , andwhen , w.h.p. and are , while

is . Besides, Bettstetter et al. [23] examine theimpact of randomized beamforming on multihop wireless net-works via simulations.

However, these researches have not taken the transmissionand the reception schemes into consideration, which are actu-ally very important when we discuss the connectivity problem.Similar to that in [24] where capacity of wireless networksusing directional antennas is discussed, we classify in this paperWANETs using directional antennas into four types accordingto their transmission and reception schemes, i.e., DTDR net-works, DTOR networks, OTDR networks, and OTOR networks.We study the CTR in these networks based on a more realisticdirectional antenna model with both main lobe antenna gain

1The chromatic number, the directed clique number, and the undirected cliquenumber of a graph � are the least number of colors needed to color the graph�, the size of the maximum directed clique in graph �, and the size of themaximum undirected clique in graph �, respectively.



Fig. 1. Our switched beam directional antenna model.

and non-negligible side lobe antenna gain. Then, for each kindof network, we formulate a non-linear programming problemto find the optimal antenna pattern , given the numberof beams and the path loss exponent , in order to provideasymptotic network connectivity with the minimum CTP. Wealso examine the impact of and on network connectivity.

III. PRELIMINARIES

A. Directional Antenna Model

Denote a vector in three-dimensional space by ,then the gain of an antenna in the direction is given by [25]

where is the power density in the direction is theaverage power density over all directions, and isthe efficiency of the antenna which accounts for losses. Clearly,we can see that an omnidirecitonal antenna has a gain of 0 dBand a directional antenna can have a higher gain than that. Due tothe higher gain and less interference when it is beamforming ina specific constrained direction, a directional antenna can offera longer transmission distance than an omnidirectional antenna.

In the current literature, there are three primary types of direc-tional antenna systems: the switched beam antenna system, thesteered beam antenna system, and the adaptive antenna system[26]. In this study, we use the switched beam antenna system,which consists of several highly directive, fixed, pre-definedbeams and each transmission uses only one of the beams. Ourstudy assumes that every antenna has beams exclu-sively and collectively covering all directions. The main lobeantenna gain is denoted by and the side lobe antenna gainis denoted by . We assume and are constants in themain lobe direction and side lobe directions, respectively. Onesuch example with four beam directions is shown in Fig. 1.Moreover, directional antennas work either in the directionalmode or in the omnidirectional mode

.Let be the transmission power, and the surface area of

the sphere with center at the transmitter and radius . As shownin Fig. 2, the surface area on the sphere for a beamwidth of

is , where is , and is . By the

Fig. 2. Illustration for calculating the main lobe and the side lobe antenna gain.

Fig. 3. The main lobe antenna gain with � � �� , and � � �� or� � ��.

definition of antenna gain, when we neglect the side lobe gain, we have [27]

and when we consider the side lobe gain , we have

(1)

where and are the main lobe directional antenna gainand the side lobe directional antenna gain, respectively. Since

, (1) can be simplified as

(2)

from which we can see that there is a relationship among themain lobe antenna gain , the side lobe antenna gain , andthe number of beam directions , which is related to the beamwidth by . Some of the values derived according tothese equations by setting are shown in Fig. 3.



B. Power Propagation Models

Power propagation models are used to predict the receivedsignal strength. A general model is given in (3):

(3)

where and are the transmission power and the receptionpower, respectively, and are the gain factors for the trans-mitter antenna and the receiver antenna, respectively, andare the antenna heights of a transmitter and a receiver, respec-tively, is the distance between the transmitter and the receiver,

is the system loss factor not related to propagationis the wavelength, is a function, and is the path loss

exponent.Two specific models are shown in the following [28], with the

path loss exponent equal to 2, and 4, respectively.The free space propagation model, which is used when the

transmitter and receiver have a clear, unobstructed line-of-sightpath between them, is given below:

The two-ray ground reflection model, which considers both thedirect path and a ground reflected path between the transmitterand the receiver, is as follows:

From the above, we can see that the path loss exponent canbe used to characterize the propagation environment. Besides,the value of usually ranges from 2 to 5 in outdoor environ-ments [28].

IV. NECESSARY AND SUFFICIENT CONDITIONS FOR

ASYMPTOTIC CONNECTIVITY

In this section we present the necessary and sufficient condi-tions for achieving asymptotic connectivity in WANETs usingdirectional antennas. As we mentioned earlier, there are four cat-egories of WANETs in terms of the transmission and the recep-tion schemes: DTDR networks, DTOR networks, OTDR net-works, and OTOR networks. We will discuss the connectivityproblem in the first three kinds of networks, respectively, in thefollowing three subsections. The case in OTOR networks is thesame as that discussed in [14].

We first give the assumptions we use in this paper:(A1) There are static nodes uniformly and independently

distributed in a disk of unit area on the plane.(A2) All nodes are equipped with the same switched beam

directional antennas. The number of beams is , the main lobeantenna gain is , and the side lobe antenna gain is .

(A3) All nodes have the same transmission power and trans-mission range.

(A4) Each node randomly beamforms in one of the direc-tions, with a probability of .

(A5) Edge effects are neglected.

We denote the resulting network graph by , whereis the vertex set, and is the edge set defined by a func-

tion : . Notice that depends on the transmissionand reception schemes. So we use in DTDR networks, inDTOR networks and in OTDR networks.

A. DTDR Networks

We first derive the necessary condition for achieving asymp-totic connectivity in DTDR networks. Let denote the omni-directional transmission range with the transmission power .With the same transmission power, let denote the trans-mission range when two nodes beamform to each other, thetransmission range when only one of the two nodes beamformsto the other, and the transmission range when neither of thetwo nodes beamforms to the other. According to the power prop-agation model in (3), we have

(4)

(5)

(6)

Fig. 4 shows the communication area of an arbitrary nodein DTDR networks. Let denote the distance to node , Area Iand Area II the areas where andin the main lobe direction, respectively, and Area III and AreaIV the areas where and in the sidelobe directions, respectively. We observe that:

(1) Each node can communicate with the neighbors in AreaI with probability , where

(2) Each node can communicate with the neighbors in AreaII with probability , where

(3) Each node can communicate with the neighbors in AreaIII with probability , where

(4) Each node can communicate with the neighbors in AreaIV with probability , where

Thus, the effective area of nodes in DTDR networks, denotedby , is



Fig. 4. The communication area of a node � in DTDR networks when direc-tional antennas have four beam directions.

Let . Then, .Denote the probability that two nodes, and , which also de-

note the positions of these two nodes in , in DTDR networksare connected by . As shown in Fig. 4, we notice that

and are correlated, where is another neigh-boring node of . So, it is very difficult and complex to directlyderive analytically. However, as pointed out in [29], wecan approximate the connectedness function of an anisotropicsystem simply by that of an spherical reference system denotedby , i.e.,

(7)

where denotes the distance between two nodes and ,the angular brackets indicate an orientation average. Note that[29] shows this approximation is exact when the system is ei-ther slightly anisotropic or very anisotropic. Besides, when thesystem is neither slightly anisotropic nor very anisotropic, thisformalism is still capable of describing the connectivity of ran-domly distributed particle systems over a wide range of particleanisotropy. Thus, in our case, we can use to determine theedge set in the graph , especially when the beamnumber goes large.2

In the following, we use to replace indicating thatthe transmission range is dependent on the number of nodes

in order to achieve connectivity. We also denote the proba-bility that is disconnected by . Guptaand Kumar has shown [14] that in networks with omnidirec-tional antennas, if , then

2As we will see later, directional antennas do need to have very small sidelobes in order to achieve connectivity with low transmission power. In particular,the side lobe gain� needs to approach 0 to have optimum pattern as the beamnumber � increases.

when . Similarly, we can have thefollowing result.

Theorem 1: (i) In DTDR networks, if, then

where .(ii) In DTDR networks, if and

, then is asymptoti-cally disconnected with positive probability.

Next, we derive the sufficient condition for achieving asymp-totic connectivity in DTDR networks using a different approachfrom that in [14]. We will need some results from continuumpercolation by Penrose [30], [31]. Consider a graph wherenodes are distributed according to a homogeneous Poissonprocess in with intensity . We denote this graph by

, where is the vertex set and is theedge set defined by function shown in (7). In addition tothat, we place a node at the origin . Then, the resulting pointprocess is a Poisson process “conditioned to have a point atin the sense of Palm measures” [30]. We denote this new graphby , where .

We define the connected components ofas clusters. Let denote the probability that thecluster containing the origin has nodes. The percolationprobability, denoted by , is the probability thatlies in an infinite cluster when , i.e.,

Since the function satisfies forand , then, we have the following

two results, which will be used later.Lemma 1: (Theorem 3 in [30]): In the graph

,

This means that as , almost surely the origin lies ineither an infinite-order cluster or an order-1 cluster (i.e., it isisolated).

Lemma 2: (Theorem 6.3 in [32]): When , there isat most one infinite cluster in .

Based on Lemma 1 and 2, the following result can be easilyobtained.

Lemma 3: As , the probability that the graphis connected is asymptotically the same

as the probability that the graph has noisolated nodes, i.e.,



In [30], it has been shown that

Let . If , then we have. So, from (8), the

probability of the origin to be isolated is:

Let be the expected number of order-1 cluster andthe probability that there is at least one order-1 cluster

in graph . Then,

So, if , thenas .

Thus, from Lemma 3, we can obtain

Since when the number of nodes is sufficiently large, the dif-ference between and is neg-ligible [14], then Theorem 2 directly follows.

Theorem 2: In DTDR networks, ifand , then the graph is asymptoticallyconnected with probability 1.

Combining Theorem 1 and Theorem 2, we arrive at our firstmain result.

Theorem 3: In DTDR networks, , with, is connected with probability 1 as

if and only if .

B. DTOR Networks

In this subsection, we derive the necessary and sufficientcondition for achieving asymptotic connectivity in DTORnetworks. Once again, let and denote the transmissionranges, respectively, when the transmitter beamforms towardthe receiver with the main lobe gain and the side lobe gain

, respectively. By the model in (3), we can obtain

(9)

(10)

Fig. 5 shows the communication area of an arbitrary nodein DTOR networks. Let denote the distance to node , Area I

Fig. 5. The communication area of a node � in DTOR networks when direc-tional antennas have four beam directions.

and Area II the areas where and inthe main lobe direction, respectively, and Area III and Area IVthe areas where and in the side lobedirections, respectively. We observe that

1) Each node can communicate with the nodes in AreaI with probability of . Recall that inSection IV.A, the communication is bidirectionallysymmetric, i.e., if node A can communicate with node B,then node B can also communicate with node A. However,in DTOR networks, the communication is bidirectionallyasymmetric, i.e., if node A can communicate with nodeB, B may not necessarily be able to communicate withA. For example, if a node B is in Area I of node A, A isbeamforming to B, but B is not beamforming toward A,then A can send packets to B but B cannot send packetsto A. Specifically speaking, we define that if two nodescannot be connected in any direction, the connectivitylevel is 0; and that if two nodes can be connected only inone direction, the connectivity level is 0.5; and that if twonodes can be connected in both directions, the connectivitylevel is 1. Thus we have

2) Each node can communicate with the nodes in Area IIwith probability of , where

3) Each node can communicate with the nodes in Area IIIwith probability of , where

4) Each node can communicate with the nodes in Area IVwith probability of , where



Thus, the effective area of nodes in DTOR networks is

Let . Then, .Following the procedures in Section IV.A, we can also useto determine the edge set in the graph , where

Thus, we can obtain the following results.Theorem 4: In DTOR networks, if

and , then is asymptot-ically disconnected with positive probability.

Theorem 5: In DTOR networks, ifand , then the graph is asymptoticallyconnected with probability 1.

In summary, we have the result below for the DTOR net-works.

Theorem 6: In DTOR networks, , with, is connected with probability 1 as

if and only if .

C. OTDR Networks

In OTDR networks, the connection function isthe same as the connection function in DTORnetworks. So the effective area of a node in OTDRnetworks is the same as that in DTOR networks, i.e.,

. Then, let, which is the same as , and

we only present the result here.Theorem 7: In OTDR networks, , with

, is connected with probability 1 asif and only if .

V. MINIMIZING THE CRITICAL TRANSMISSION POWER

In Section IV, we have given the necessary and sufficientconditions for networks using directional antennas to achieveasymptotic connectivity, i.e.,

(11)

where in DTDR networks,and in DTOR and OTDRnetworks, as shown in Theorem 3, 6 and 7, respectively.

In addition, in [14], Gupta and Kumar show that when omni-directional antennas are used, the necessary and sufficient con-ditions for OTOR networks to achieve asymptotic connectivityis

(12)

Comparing (11) with (12), we have

(13)

where is the critical transmission range in OTOR networks,and are the critical transmission ranges inDTDR, DTOR and OTDR networks, respectively. We observethat if are greater than 1, the critical transmis-sion ranges when directional antennas are used will be smallerthan that when omnidirectional antennas are used.

Assume that the reception power needs to be greater than aconstant threshold in order for the receiver to correctlyreceive the message. According to the power propagation modelintroduced in Section III.B, we have

where is a constant.Let , and denote the critical transmission

powers in OTOR networks, DTDR networks, DTOR networksand OTDR networks, respectively. Then,

(14)

where . So, in order to save power when using di-rectional antennas, our objective is to minimize , respectively,for , which is equivalent to maximizing , re-spectively, for , as shown below.

(15)

where, and .

Note that the first constraint is derived based on (2) by noticing, and the second one is due to the characteristics of

directional antennas as introduced in Section III.A. We haveseveral cases:

(1) When , we have and . Then,for , we obtain

according to Holder’s inequality. Thus,

So, when , all the maximum values can be achieved onlywhen , i.e., when directional antennas work inthe omni-directional mode. When , it is obvious that themaximum values of and are all 1, and they can beachieved for any . As a result, when ,the minimum critical transmission powers in DTDR, DTOR andOTDR networks are the same as the critical transmission powerin OTOR networks.



TABLE ISOME CALCULATED VALUE OF ��

(2) When , since, the three optimization problems presented in (15)

can all be formulated as the same non-linear programming asshown below:

(16)

where .In wireless networks, the path loss exponent is determined

by the environment, so it can be considered as a known factor.Besides, the number of beams of a directional antenna is aconstant integer greater than 1. Thus, for each value of and

, we can find optimal values of and to maximize.

By setting to 3, which is usually the case in urban areas[28], we calculate some “typical” values ofwhen the number of beams is 4, 6, and 8, respectively, asshown in Table I. Here, “typical” is referred to the settings where

dBi, since usually we want to restrain the side lobeantenna gain in order to increase the main directional transmis-sion range. However, this is not necessarily true if we want toachieve connectivity with a smaller transmission power.

From Table I, we can see that “typically” using directional an-tennas results in smaller critical transmission ranges, and hencesmaller critical transmission powers than using omnidirectionalantennas according to (13) and (14) since

. From this table, we can also observe that the antenna patternhas great impacts on the value of , and in turnthe network connectivity.

So far, we have shown how to maximize ,with the maximum value denoted by , givena certain value of and , respectively. Some values of

Fig. 6. The values of �� for � � �� , and � � �� .

Fig. 7. The impact of path loss exponent on �� .

when and have alreadybeen shown in Table I. In the following, we show some moreresults of .

The values of for andare shown in Fig. 6. Fig. 7 shows more clearly the impact

of the path loss exponent on . We observe thatthe value of can be approximated by a powerfunction , where , and

This shows that, with fixed increasesas increases, while with fixed de-creases as increases. The latter case is intuitively truebecause the increase of implies the power propagationenvironment is getting worse and it will be more diffi-cult to achieve network connectivity. Besides, we alsofind that when and when

, for all .Furthermore, the value of the side lobe antenna gain for

to achieve the maximum is shown in Fig. 8.We observe that in order for to be large, the



Fig. 8. The values of � to achieve �� .

side lobe gain has to approach 0 and the main lobe gainhas to approach .

In addition, for any , we have

which means that with the same parameter set ,if the critical transmission range in DTOR or OTDR networksis smaller than that in OTOR networks, then the critical trans-mission range in DTDR networks is further smaller than thatin DTOR and OTDR networks, and vice versa. In other words,with the same parameter set , if the criticaltransmission power in DTOR or OTDR networks is smallerthan that in OTOR networks, then we can save even more powerin DTDR networks, and vice versa.

As a result, we find that with the same number of beamsand the same path loss exponent ,

the minimum critical transmission power in DTDR networks issmaller than that in DTOR and OTDR networks, which is fur-ther smaller than that in OTOR networks. However, when thenumber of beams is equal to 2, then with the same path lossexponent , the minimum critical transmission power in DTDR,DTOR, and OTDR networks are all the same, which are equalto that in OTOR networks.

VI. SOME INSIGHTS ON OUR RESULTS

Recall that Gupta and Kumar [14] conclude that when om-nidirectional antennas are used, with

is connected with probability 1 as if andonly if . This means that the expected number ofneighbors of a node, which is , i.e., , has toapproach infinity in order to achieve connectivity as .We define as the critical expected number of neighborswhich is directly determined by critical transmission power.

In this paper, we have already shown that when directionalantennas are used, with ,

where is equal to 1, 2, or 3, corresponding to the cases ofDTDR, DTOR or OTDR networks, respectively, are connectedwith probability 1 as if and only if .This implies that in order to achieve connectivity, the expectednumber of neighbors of a node using directional antennas, whichis , i.e., , still needs to approach infinityas . However, the critical expected number of neigh-bors of a node, which is equal to as defined before,i.e., , can be just if can be on the order of

. Therefore, when using directional antennas, we can savemore transmission power if we can choose larger . This is alsoshown clearly in (14).

In Section V, we show that increases asincreases in the range of [2, 1000]. Does keepincreasing when ? We will try to answer this questionby looking into the problem of maximizingwith respect to the number of beams to see how large

can be, i.e.,

Since increases as either increasesor increases. So the maximum of canbe achieved only when , where

, the same as that defined in (15).Thus, can be represented by , where

(17)

1) When , we have

As goes large, we obtain

(18)

since and .So, , which can be

achieved when , and . Thus, we have

2) When , with and fixed, isachieved when . Let denote the result bysolving the equation, then we obtain



(19)

where for any.

Substituting (19) into (17), we obtain

Then, we obtain that for ,

which is achieved when , where .3) When or , according to (17), we

obtain

As a result, when or , we have

as . However, when , we obtain

which is achieved when , and .Moreover, when ,

which is achieved when , and .Since

, from the above, we obtain that for, respectively,

In conclusion, we show that for anycan be made very large by adjusting the directional an-

tenna pattern, i.e., , when is large. This means thatif with some transmission power, the critical expected numberof neighbors of a node is only by using omnidirectionalantennas, we can still make the network connected by using di-rectional antennas with the same transmission power. Thus, wecan save a lot of power with directional antennas because inOTOR networks, the transmission power needs to be set suchthat each node has expected neighboring nodes. Be-sides, for , the maximum values ofare in the range . Thus, we can still save power by usingdirectional antennas.

VII. AN EXTENDED CASE

In Section IV, we derive the necessary and sufficient condi-tions for achieving asymptotic connectivity with the assump-tion that each node in the network randomly beamforms in oneof the beam directions. However, if there exists a scheduleaccording to which two neighboring nodes can be guaranteedto beamform to each other to carry out the communication, theconnectivity problem would be rather different. In this section,we discuss this new connectivity problem in details.

We still classify WANETs using directional antennas intofour categories, i.e., DTDR networks, DTOR networks, OTDRnetworks, and OTOR networks. Then, the effective areas ofDTDR, DTOR, and OTDR networks, denoted by ,and , respectively, are equal to:

where and are defined in (4) and (9), respectively.Following the process in Section IV, we obtain the following

result.Theorem 8: If there exists a schedule according to which two

neighboring nodes can be guaranteed to beamform to each otherto carry out the transmissions, for DTDR, DTOR, and OTDRnetworks, with respectively,are connected with probability 1 as if and only if

, where .Note that , and are larger than , and , respec-

tively, which means the critical transmission ranges are smallerthan those in previous case, and so are the critical transmissionpowers.

As shown in (15), we havewhere . Thus,

As a result, as for ,respectively, which is achieved when , and .This means, for any , if with a transmission powerlevel so that the critical expected number of neighbors of a nodeis only when using omnidirectional antennas, we can stillmake the network connected by using directional antennas withthe same transmission power.

Besides, we also notice that at current stage of technologythe beam number is very small. Can our claim that the crit-ical number of neighbors can be just to achieve connec-tivity still hold? Denote in (16), which is con-tained in Theorem 3, 6, and 7, by , and in Theorem8 by . In order for our conclusion to hold, we can makeand be in DTOR and OTDR networks, and inDTDR networks, where is the number of nodes in the net-work. We show the values of , and in Fig. 9 when

, and . We can clearly observe thata beam number smaller than 10 is sufficient for our conclusion



Fig. 9. The values of �� and �� compared to �� when � � 100, 1000,10000, and � � 2.

to hold even in a large wireless network with 10000 nodes. Fur-thermore, we realize that as the number of nodes goes larger,we need to increase accordingly because is a monotoni-cally increasing function of . Thus, for large , our claim canonly hold if the beam number is also made large, which wehope to be true as the directional antenna technology progresses.

VIII. CONCLUSION

In this paper, we study the connectivity problem in WANETsusing directional antennas. Since transmission and receptionschemes have significant impacts on the network connectivity,we classify the networks into four categories: DTDR networks,DTOR networks, OTDR networks, and OTOR networks.

Under the assumption that each node in the network randomlybeamforms in one of the beam directions, in outdoor envi-ronments with the path loss exponent , we obtain thefollowing conclusions:

(C1) When the beam number is 2, the minimum criticaltransmission powers in DTDR, DTOR, and OTDR networks areall the same, which are equal to the critical transmission powerin OTOR networks.

(C2) When the number of beams is larger than 2, the min-imum critical transmission power in DTDR networks is smallerthan that in DTOR and OTDR networks, which is further smallerthan that in OTOR networks.

(C3) In DTDR, DTOR and OTDR networks, we can stillmake the network connected by using directional antennas evenwhen the critical expected number of neighbors is just .

Moreover, we show that (C3) actually holds for any.

Besides, we also study the connectivity problem when thereexists a schedule according to which two neighboring nodescan be guaranteed to beamform to each other to communicate.In this case, we show that (C3) actually holds for any

.Finally, we realize that in real deployment, the number of

nodes is usually limited, and hence the reliable estimation on

critical transmission range in the case with limited number ofnodes is also interesting and challenging. We will investigatethis problem in our future research.

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Pan Li (S’06) received the B.E. degree in electricalengineering from Huazhong University of Scienceand Technology, Wuhan, China, in 2005. He isworking towards the Ph.D. degree at the Departmentof Electrical and Computer Engineering, Universityof Florida, Gainesville.

His research interests include capacity and connec-tivity analysis, medium access control, routing algo-rithms, and cross-layer design in wireless networks.He is a student member of the IEEE and the ACM.

Chi Zhang (S’06) received the B.E. and M.E.degrees in electrical engineering from HuazhongUniversity of Science and Technology, Wuhan,China, in July 1999 and January 2002, respectively.Since September 2004, he has been working towardsthe Ph.D. degree at the Department of Electricaland Computer Engineering, University of Florida,Gainesville.

His research interests are network and distributedsystem security, wireless networking, and mobilecomputing, with emphasis on mobile ad hoc net-

works, wireless sensor networks, wireless mesh networks, and heterogeneouswired/wireless networks.

Yuguang Michael Fang (S’92–M’97–SM’99–F’08)received the Ph.D. degree in systems engineeringfrom Case Western Reserve University in January1994 and the Ph.D. degree in electrical engineeringfrom Boston University in May 1997.

He was an Assistant Professor in the Departmentof Electrical and Computer Engineering, New JerseyInstitute of Technology, from July 1998 to May2000. He then joined the Department of Electricaland Computer Engineering at the University ofFlorida in May 2000 as an Assistant Professor, got

an early promotion to an Associate Professor with tenure in August 2003 andto a full Professor in August 2005. He held a University of Florida ResearchFoundation (UFRF) Professorship from 2006 to 2009. He has published over200 papers in refereed professional journals and conferences.

Dr. Fang received the National Science Foundation Faculty Early CareerAward in 2001 and the Office of Naval Research Young Investigator Award in2002. He is the recipient of the Best Paper Award in IEEE Int. Conf. NetworkProtocols (ICNP) in 2006 and the recipient of the IEEE TCGN Best PaperAward in the IEEE High-Speed Networks Symposium, IEEE Globecom in2002. He is also active in professional activities. He is an Fellow of IEEEand a member of ACM. He has served on several editorial boards of technicaljournals including IEEE Transactions on Communications, IEEE Transactionson Wireless Communications, IEEE Transactions on Mobile Computing andACM Wireless Networks. He has been actively participating in professionalconference organizations such as serving as the Steering Committee Co-Chairfor QShine, the Technical Program Vice-Chair for IEEE INFOCOM’2005,Technical Program Symposium Co-Chair for IEEE Globecom’2004, and amember of Technical Program Committee for IEEE INFOCOM (1998, 2000,2003–2009).


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