Collision-Tolerant Transmission with Narrow-beam Antennas

doi:10.4236/ijcns.2008.14035

I. J. Communications, Network and System Sciences, 2008, 4, 285-385

Published Online November 2008 in SciRes (http://www.SciRP.org/journal/ijcns/).

Collision-Tolerant Transmission with

Narrow-Beam Antennas

Hong-Ning DAI1, Kam-Wing NG1, Min-You WU2, Bo LI3

1Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong SAR

2Computer Science and Engineering Department, Shanghai Jiao Tong University, Shanghai, P.R.China

3School of Electronic and Information Engineering, South China University of Technology, Guangzhou, P.R.China

Email: {hndai,kwng}@cse.cuhk.edu.hk, wu-my@sjtu.edu.cn, bill.leebo@gmail.com

Received April 29, 2008; revised September 1, 2008; accepted September 23, 2008.

Abstract

The application of directional antennas in wireless ad hoc networks brings numerous benefits, such as

increased spatial reuse and mitigated interference. Most MAC protocols with directional antennas are based

on the RTS/CTS mechanism which works well in wireless ad hoc networks using omni-directional antennas.

However, RTS/CTS frames cannot mitigate the interference completely. Besides, they also contribute a lot

to the performance overhead. This paper studies the problem from a new perspective. We have found that

the transmission success probability under directional transmission and directional reception is quite high

when the antenna beamwidth is quite narrow. Motivated by the analytical results, we design a lightweight

MAC protocol without RTS/CTS frames. The evaluation results demonstrate that this new protocol

performs better than MAC protocols based on the RTS/CTS mechanism. The results also show that a

collision-tolerant transmission is feasible under the narrow beam configuration.

Keywords: Wireless Networks, Directional Antennas, Medium Access Control

1. Introduction

The application of directional antennas to wireless ad

hoc networks has received enormous interest in recent

years. Directional antennas can greatly improve network

performance by increasing network connectivity, expanding

transmission range, enhancing spatial reuse and reducing

interference. Recent studies such as [1–10] focus on

designing new MAC layer protocols to improve network

performance.

Most of these MAC schemes with directional antennas

are based on a four-way handshaking scheme, known as

request-to-send/clear-to-send (RTS/CTS). The RTS/CTS

mechanism has been proposed to resolve the hidden

terminal problem in wireless networks using omni-directional

antennas which can broadcast RTS/CTS frames to inform

neighboring nodes of the oncoming transmission. Those

nodes that have received the RTS/CTS frames can defer

their transmission to avoid collisions. However, using

RTS/CTS cannot eliminate hidden terminals completely

even in wireless networks with omni-directional antennas

[11]. Furthermore, Choudhury et al. [5] have found that

using directional antennas causes new interference such

as new hidden terminals and the deafness problem, which

cannot be solved by using the RTS/CTS mechanism.

Essentially, directional antennas can radiate or receive

signals more effectively in one direction, which can cause

much less interference than omni-directional antennas. So,

does the RTS/CTS mechanism still work well with

directional antennas?

Many novel mechanisms have been proposed to

eliminate the new hidden terminal problem and the

deafness problem, which are caused by directional

antennas. Although Korakis et al. [3] propose a Circular-

DMAC scheme to combat the new hidden terminal

problem and the deafness problem, transmitting multiple

RTS/CTS frames for each data transmission severely

degrades the performance. Other schemes, such as Tone-

based DMAC [7] and BTDMAC [12] can alleviate the

impacts of the hidden terminal and deafness problems by

sending tones over another channel or over the data

channel after data transmission. However, these bulky

and complicated schemes also bring additional cost and

286 H. N. DAI ET AL.

performance penalty.

How to use directional antennas in wireless networks

more effectively? We address this problem from another

viewpoint. When the beamwidth of a directional antenna

is lessened (a narrower beamwidth), the interference caused

by the antenna will also be reduced. We have found that

when the beamwidth is quite narrow and the network is

not so dense, the collision probability is quite low. It is

the purpose of this paper to study the performance of

wireless networks using narrow-beam antennas. In parti-

cular, we are interested in the following problems:

♦ What will happen when the beamwidth of the

directional antennas is lessened? What is the impact of

other factors on the success transmission probability,

such as the node density?

♦ How effective is the RTS/CTS mechanism in wireless

networks using directional antennas? If RTS/CTS is

turned off, will the network throughput degrade

significantly?

In the next section, we briefly survey the related work

in the literature. Section 3 describes the models used in

this paper and analyzes the success transmission proba-

bility for directional transmission and directional reception.

In Section 4, we present a lightweight MAC protocol

without the RTS/CTS mechanism and compare its per-

formance with a representative MAC protocol using the

RTS/CTS mechanism. Section 5 offers some implications

of our results. Finally, we summarize our paper in

Section 6.

2. Related Work

Many studies [1–10] focus on designing new MAC

protocols with directional antennas. Most of them are

based on the IEEE 802.11MAC [13], which typically

uses RTS/CTS to reduce interference in wireless networks.

Although the RTS/CTS mechanism works well in wireless

networks equipped with omni-directional antennas, it

cannot mitigate interference completely [11]. Besides,

using RTS/CTS in wireless networks with directional

antennas is not as effective as we expected. For example,

ref. [5] shows that RTS/CTS cannot completely mitigate

new interfering nodes caused by directional antennas.

To address the new hidden terminal problem and the

deafness problem, many researchers propose more

complex schemes, such as Circular-DMAC [3], Tone-

based DMAC [7] and BT-DMAC [12]. Although they

can mitigate the impacts of hidden terminals and deafness,

they also bring additional cost on network performance.

For example, Circular-DMAC needs a sender to transmit

multiple RTS frames before each data transmission,

which greatly degrades the network performance. Tone-

based DMAC and BT-DMAC also need to send out-of-

band tone signals to reduce interference.

Other studies [14–17] concentrate on capacity analysis

and performance evaluation on wireless ad hoc networks

using directional antennas. Yi et al. [15] have found that

using directional antenna in arbitrary networks achieves a

capacity gain of 2 /

π αβ

when both transmission and

reception are directional. Here,

α

and

β

are transmitter

and receiver antenna beamwidths, respectively. Under

random networks, the throughput improvement factor is

2

4/ ()

π αβ

for directional transmission directional

reception. Ref. [14] studies the asymptotic bounds on the

amount of capacity gains that directional antennas can

acquire. Wang et al. [16] model and analyze multiple

directional transmission and reception modes coupled

with omni-directional or directional receptions. Carvalho

and Garcia-Luna-Aceves [17] propose a realistic

analytical model which considers the binary exponential

back-off operation of IEEE 802.11.

In this paper, we try to find the relationship between

the interference, the beamwidth of directional antennas

and the density of nodes.

3. Analytical Models

In this section, we analyze the successful transmission

probability with directional antennas. The successful

transmission probability does not only depend on the

activity of the interfering nodes but also on the

transmission/reception mode of directional antennas.

First, we present the antenna model in Section 3.1.

Section 3.2 discusses the interference range for

directional transmission. Finally, we analyze the

successful transmission probability under the directional

transmission and directional reception mode.

3.1. Antenna Model

The radiation pattern of a direction antenna is often

depicted as the gain values in each direction in space. We

can project the radiation pattern of an antenna to an

azimuthal or elevation plane. The projection of the

pattern typically has a main lobe (beam) of the peak gain

and side-/back-lobes of smaller gains.

Since modeling a real antenna with precise values for

main and side-/back-lobes is difficult, we use an approximate

Figure 1. The antenna model.

COLLISION-TOLERANT TRANSMISSION WITH NARROW-BEAM ANTENNAS 287

antenna pattern [18]. In an azimuthal plane, the main lobe

of antenna can be depicted as a sector with angle

θ

,

which is denoted as the main beamwidth of the antenna.

The side-/back-lobes are aggregated to a circle, as shown

in Figure 1. The narrower the main beamwidth of the

antenna is, the smaller the side-/back-lobes are. Take the

above antenna model as the example. The gain of the

main beam is more than 100 times of the gain of side-

lobes when the main beamwidth is less than 40° [18].

Thus, the side-/back-lobes can be ignored when the main

beam is quite narrow.

Our proposed model assumes that a directional

antenna gain

d

G

is within a specific angle

θ

, where

θ

is

the beamwidth of the antenna. The gain outside the

beamwidth is assumed to be zero. At any time, the

antenna beam can only be pointed to a certain direction,

as shown in Figure 1, in which the antenna is pointing to

the right. Thus, the probability that the beam is switched

to cover each direction is

/ (2)

θ π

. The antenna gain

pattern is given by:

( )d

G if angle within

g

0 otherwise

θ



=



3.2. Interference Range

Compared with omni-directional antennas, directional

antennas have different transmission properties.

Directional antennas can radiate or receive ratio signals

more effectively in a certain direction than other

directions. Thus, directional antennas have different

transmission range and interference region, compared

with omni-directional antennas. In this subsection, we

investigate the interference region of directional antennas

and the relationship between the transmission range and

the interference range.

When a signal is propagated from the transmitter to

the receiver, whether it is correctly accepted by the

receiver is mainly determined by the receiving power of

the signal at the receiver end. In open space, if the

transmitting power is fixed, the receiving power is mostly

decided by the path loss along the distance between the

transmitter and the receiver. Under this condition, multi-

path and shadowing effects can be ignored since they are

so trivial compared with the large path loss. Therefore, in

this paper, we assume that the signal propagation follows

the two-way ground model which is typically used in

open space.

According to [19], under the assumption of the two-

way ground model, the receiving power of a signal at the

receiver can be calculated by the following equation.

2 2

4

( )

t r

rt tr

h h

PdPG G

d

= (1)

where

( )

r

P d

is the receiving power at the receiver which

is far from the transmitter with the distance

d

,

t

P

is the

Figure 2. The interference model.

transmitter power,

t

G

and

r

G

are the transmitter antenna

gain and the receiver antenna gain, respectively,

and

t

h

and

r

h

are the antenna height of the transmitter and

the antenna height of the receiver, respectively.

Consider a large-scale wireless ad hoc network with

n

static nodes. Without loss of generality, the distribution

of the nodes follows a Poisson distribution with a

parameter

ρ

over the 2-D plane. The probability

( ,)

p iS

of finding

i

nodes in an area of S is given by:

( )

( ,)!

i

S

p iSe

i

ρ

−

= (2)

We also assume that every node has an identical

antenna and transmits with a fixed power. Thus, each

node has the same transmitting range

t

R

and the same

interference range

i

R

. In the scenario shown in Figure 2,

suppose that node

i

X

transmits to node

j

X

over a

channel. The receiver

j

X

locates exactly within the

transmitting range

t

R

of the transmitter

i

X

.

The successful reception of the signal is mainly

decided by the signal-to-interference-plus-noise ratio

(SINR), which is often required to be greater than a

threshold. When their transmission is on-going, an

interfering node

k

X

at the interference range

i

R

away

from the receiver starts the transmission toward the

receiver at the same time. So, it will have an interfering

signal with the strength

i

P

at the receiver

j

X

. Since the

thermal noise is negligible compared with interference

signals, similar to [11], we do not count it in our model

as well. Thus, we have4

4

i

r

t t

R

P

SINR PR

σ

= =≥

, where

σ

is

the SINR threshold. In practice,

σ

is usually set to 10.

So, we get the interference range4

i t

R R

σ

=.

3.3. Directional Transmission and Directional

Reception

A directional antenna has two modes: an omni-

directional mode with a gain

o

G

and a directional mode

with a gain

d

G

. Since antennas in the directional mode

can radiate or receive radio waves more effectively in

Xj

Rt

Ri

Xi

Xk

θ

288 H. N. DAI ET AL.

some directions than in others, the directional gain

d

G

is

generally greater than the omni-directional gain

o

G

. The

transmitter or the receiver equipped with a directional

antenna can choose any one of the two modes to transmit

or receive frames. Hence, there are four combinations for

the transmission and reception modes of directional

antennas: 1) Omni-directional Transmission and Omni-

directional Reception (OTOR); 2) Directional

Transmission and Omni-directional Reception (DTOR);

3) Omni-directional Transmission and Directional

Reception (OTDR); 4) Directional Transmission and

Directional Reception (DTDR).

According to Equation (1), the larger the antenna

gains both the transmitter and the receiver have, the

higher the receiving power the receiver has. Besides, the

transmission range between the transmitter and the

receiver will be extended if the antenna gains of them are

increased. Thus, when both the receiver and the

transmitter use the directional mode, the communication

range between them is maximized. On the other hand, the

receiver is only susceptible to the interfering signals from

its receiving direction when it is using the directional

mode. So, DTDR also has the smallest interference area

compared with the other three modes. Hence, DTDR is a

preferred method to utilize directional antennas. In this

paper we only discuss the transmission under the DTDR

mode.

Let us consider the scenario shown in Figure 2. When

node

i

X

begins to transmit with node

j

X

, this packet is

successfully received by node

j

X

if no node within the

sector region covered by

j

X

’s antenna beam transmits

toward

j

X

. First, we need to calculate the probability that

no node can interfere with node

j

X

. Since the placement

of nodes follows the 2-D Poisson distribution with the

density

ρ

, there are 2

2

i

R

θ

ρπ

π

⋅

nodes within the sector

region covered by

j

X

’s antenna beam. The area of this

region is denoted byS. Among these nodes, the

interfering node

k

X

can cause interference with node

j

X

only when it has a frame to send and its antenna beam

is pointed to node

j

X

. We assume that a node begins to

transmit with a probability

p

.Then, the probability that

node

k

X

can interfere with node

j

X

is

2

p

θ

π

⋅

. There, the

probability

P

that no nodes within region can cause

collisions with node

j

X

is given by:

2 2

0

( )

2 2

( )

(1 )

2 !

i

i S

i

p SpR

S

P pe

i

e e

ρ

θ θ

ρ ρπ

π π

θ ρ

π

∞−

=

− −

= −⋅

= =

∑ (3)

To simplify the calculation, we use

2

t

N R

ρπ

=, which

denotes the average number of nodes within a node’s trans-

Figure 3. The probability of a successful transmission.

mission range. Since4

i t

R R

σ

=, we have 2

i

R N

ρπ σ

=.

Replacing the corresponding part in Equation (3), we have:

2

( )

2

p N

P e

θ

σ

π

−

= (4)

When

0.1

p

=

and

10

σ

=

, we set different

4,8,12,16,20

N

=

respectively and then we get the results

in Figure 3. Figure 3 shows that the successful

transmission probability is high when the beamwidth is

narrow. For example, when

θ

is less than

6

π

, the success

probability is always above 95%. One possible reason is

that using directional mode at the receiver end can

greatly reduce the collision probability.

Results under a narrow beamwidth (

12

π

θ

≤) are also

tabulated in Table 1, which shows that the transmission

under DTDR is less vulnerable to interference when the

beamwidth is quite narrow.

The analytical results under DTDR show that the

successful transmission probability is quite high when the

beamwidth is lessened enough. For example, when

12

π

θ

≤ (i.e., 15°), the success probability is always

above 98%. A beamwidth of 15° is a feasible angle for

most directional antennas. Thus, intuitively, there is an

Table 1. The probability of a successful transmission under

the very narrow beam.

48

π

θ

=

36

π

θ

=

24

π

θ

=

12

π

θ

=

4

N

=

0.9999 0.9998 0.9995 0.9978

8

N

=

0.9997 0.9995 0.9989 0.9956

12

N

=

0.9996 0.9993 0.9984 0.9934

16

N

=

0.9995 0.999 0.9978 0.9913

20

N

=

0.9993 0.9988 0.9973 0.9891

COLLISION-TOLERANT TRANSMISSION WITH NARROW-BEAM ANTENNAS 289

interesting question: can the transmission continue even

if there exist few collisions? In other words, when the

beamwidth of antennas is narrow enough and the

collision probability is quite low, can the transmission be

collision-tolerated?

4. Lightweight MAC Protocol

In this section, we propose a lightweight MAC scheme

denoted as Basic Directional Transmission and

Directional Reception (B-DTDR), which turns off the

control frames of request to send (RTS) and clear to send

(CTS). It has a rival termed RTC/CTS Directional

Transmission and Directional Reception (RTS-DTDR).

Then, we compare the performance of B-DTDR with that

of RTS-DTDR and discuss the implications from this

lightweight scheme.

4.1. Quick Review of RTS/CTS Mechanisms

with Directional Antennas

B-DTDR scheme keeps the basic collision avoidance

features, such as the exponential backoff and the control

frame of acknowledge (ACK). Thus, if there is a collision

with data packets, those data packets need to be

retransmitted. In IEEE 802.11 distributed coordination

function (DCF) [13], an exponential backoff scheme can

be used to avoid further collisions of data packets. At

each packet transmission, the backoff time is uniformly

chosen in a range of

( ,)

o w

where

w

is called the

contention window. At the first transmission,

w

is set to

be the minimum value of

min

W

. After each unsuccessful

transmission (no ACK received), the size of w is doubled

until it reaches the maximum value of

max

W. This

mechanism can effectively reduce the collision probability.

Most of current directional MAC schemes are using

another four-way handshaking technique which turns on

RTS/CTS frames. In stead of sending a data packet, a

transmitter sends a short control frame called request to

send (RTS). After the reception of RTS, the receiver

responds the transmitter with the frame called clear to

send (CTS). After shaking hands of RTS and CTS, the

data transmission begins. This mechanism is useful to

reduce the hidden nodes in wireless networks using

omni-directional antennas. However, it cannot mitigate

the hidden terminal problem and the deafness problem

with directional antennas [5]. In this paper, we consider a

general MAC scheme (RTSDTDR) which can be used to

represent the current directional MAC schemes since it

keeps has the main features of them. In both B-DTDR

and RTS-DTDR, RTS, CTS, data packets and ACK are

transmitted directionally.

One of the difficult problems with B-DTDR and

RTSDTDR is to find the location of a node’s neighbors,

or neighbor discovery. This problem can be solved by

using DOA caching [4] or similar mechanisms. A

specific problem with B-DTDR is how to help a receiver

to know that a transmitter is trying to send a frame to it.

Zhang [10] proposes a scheduling mechanism to address

this problem. In this paper, we assume that both B-DTDR

and RTS-DTDR can solve the neighbor discovery

problem.

On the other hand, when a node receives any frames

(RTS, CTS and data frames), it will record the

corresponding information into its DNAV (Directional

Network Allocation Vector), which is a directional

version of NAV of IEEE 802.11, proposed in [4,5].

DNAV excludes the directions and sets the corre-

sponding durations, toward which the node is not allowed

to initiate a transmission to avoid collisions with data or

control frames. When a node receives a frame and the

frame is for this node, it beamforms toward the

transmitter (switch to directional mode) and replies the

frame with a CTS (or ACK) frame. If the frames are not

for itself, it will update the sender’s information and set

the corresponding DNAVs. DNAVs are used in both B-

DTDR and RTS-DTDR.

4.2. Performance Model

In this paper, we adopt a discrete Markov chain model

used in [16, 20] to evaluate the saturation throughput and

the overhead of wireless networks (as shown in Figure 4).

We extend the model to support directional antennas.

Range extension and overhead calculation are also

considered in our model. We also adopt the assumption

that each node operate in time-slotted mode, with a time

slot

τ

. If the time slot

τ

is very small, the performance

of the time-slotted protocol is very close to that one of

the asynchronous version of the protocol [16, 20]. The

period of time during which RTS, CTS, data and ACK

frames are transmitted can be depicted as multiples of

τ

,

i.e.,

rts

t

,

cts

t

,

data

t

and

ack

t

, respectively.

The throughput is calculated by the proportion of time

that a node spends transmitting data packets successfully

on the average. Let

( )

P S

,

( )

P I

and

( )

P C

denote the

steady-state probability of SUCCESS, IDLENESS and

COLLISION, respectively. From the Markov chain model

Figure 4. The Markov chain model for a node.

290 H. N. DAI ET AL.

shown in Figure 4, we have the following equation to

calculate the throughput.

( )

()()()

data

CS I

P St

Throughput

P C TPS TPIT

⋅

=++ (5)

where

C

T

,

S

T

and

I

T

are the duration of COLLISION,

SUCCESS and IDLENESS, respectively.

The duration of time that a node stays in the

SUCCESS state,

S

T

or the collision state,

C

T

, depend on

the mechanisms of different MAC protocols. Thus, the

detailed calculation will be given in the following

subsections. The duration of a node in IDLENESS state

I

T

is

1

τ

.

Then, we need to calculate the probabilities that the

node stays in different states. From Figure 4, the steady-

state probability of IDLENESS equals:

( )()()()

II

P IP IPP SPC

=⋅ ++ (6)

Note that

()()1( )

PSP CPI

+= −, thus,

()1/ (2)

II

P IP

=− (7)

From Figure 4, the steady-state probability of

SUCCESS can be calculated by()()

IS

P SPIP

= ⋅

. Before

deriving the transition probability

IS

P

from IDLENESS to

SUCCESS, we need to calculate

( )

IS

P r

that node

i

X

successfully shakes hands with node

j

X

which is a

distance

r

way. The detailed calculation of

( )

IS

P r

will

be stated as follows.

We also derive the MAC overhead by calculating the

portion of time that a node spends transmitting control

frames on the average when data packets are successfully

transmitted.

( )

()()()

ctrl

CS I

P St

Overhead

P C TPS TPIT

⋅

=++ (8)

where

ctrl

t

is depicted as time slots which are used to

transmit control frames such as RTS, CTS and ACK.

In the following subsections, we derive the steady-

state probabilities, transition probabilities and times spent

at different states of the two MAC schemes, respectively.

4.3. RTS/CTS Based Directional Transmission

and Directional Reception (RTS-DTDR)

In this subsection, we calculate the throughput and the

overhead of RTS-DTDR. From the throughput model

presented above, we need to calculate the transition

probability

IS

P

first. Figure 5 indicates that the nodes

within the four regions (named 1, 2, 3, 4) may interfere

Figure 5. The interference region for DTDR.

with node

i

X

and node

j

X

. The transmission range

between

i

X

and

j

X

is denoted as

r

. So, the interference

range 4

i

r r

σ

=, which can be easily derived from the

results presented in Section 3.2. Since the number of

nodes depends on the area size, we need to calculate the

four areas of regions 1, 2, 3 and 4, which are denoted

as

1

S

,

2

S

,

3

S

and

4

S

, respectively:

2

1

2 2

2

2 2

3

2 2

4

/ (2)

/ (2)tan(/ 2) / 2

/ (2)/ (2)

i

S r

S rr

π θπ

π θπθ

πθ ππθ π

= ⋅

= ⋅−

=⋅− ⋅

(9)

( )

IS

P r

equals the probability that

i

X

transmits in a

given time slot, and

j

X

does not transmit in the same

time slot, and none of the nodes within the four regions

interferes with the handshake between nodes

i

X

and

j

X

.

Therefore, we have:

1 2 34

( )(1)

IS

PrppP PPP

=−⋅ ⋅⋅ ⋅

(10)

Since RTS-DTDR does not prevent interference from

neighboring nodes in regions 3 and 4, the handshake

might be interrupted at any time. Hence, the COLLISION

period

C

T

lasts from

1

rts

T t

= +

till

2

4

rts cts data ack

T tttt

=+ +++

, where one

propagation delay

τ

1is also considered.

C

T

is the mean

value of the truncated geometric distribution. Then, we

obtain the following equation to calculate

C

T

.

2 1

11

0

(1) / (1)()

T T

T Ti

Ci

Tppp Ti

−

− +

=

= −−+

∑ (11)

The probability that no nodes in region 1 interferes

with the handshake between nodes

i

X

and

j

X

is equal

to the probability that no node in this area transmits as

node

i

X

does, which can be depicted as:

1

10

2

( )

(1 )

2 !

i

S

i

p S

S

P pe

i

e

ρ

θ

ρπ

ρ

θ

π

∞−

=

−

= −⋅

=

∑ (12)

1

2

3

4

ri

r

Xi

COLLISION-TOLERANT TRANSMISSION WITH NARROW-BEAM ANTENNAS 291

The probability

2

P

is equal to the probability that no

node transmits toward node

j

X

within the duration of

time

rts

t

and no node transmits within the slot when node

i

X

begins to transmit toward node

j

X

Thus, we have

the following equation to calculate

2

P

.

2 2

( 1)

2 2

2

rts

p Stp S

P ee

θ θ

ρ ρ

π π

−+ −

=⋅ (13)

3

P

is equal to the probability that no node can interfere

with the reception of CTS and ACK frames of node

i

X

.

Hence, we have:

3 3

( 1)(1)

2 2

3

cts ack

pS tpS t

P ee

θ θ

ρ ρ

π π

−+ −+

= ⋅ (14)

In region 4, there is no interference if no node

transmits toward node

j

X

when node

i

X

is sending a data

frame. Then, we get:

4 4

( 1)

2 2

4

data

pSpS t

P ee

θ θ

ρ ρ

π π

− −+

= ⋅ (15)

Because each transmitter can choose its receiver with

the equal probability and the average number of nodes

within a region of radius

r

is proportional to

2

r

, the

probability density is the function of distance

r

between

nodes

i

X

and

j

X

, i.e.,

( )2

f rr

=, where 0

t

r R

< <

.

Therefore,

IS

P

is equal to:

0

1 2 3 4

0

( )( )

(1 )2

t

R

IS IS

R

PPrfrdr

ppP PPPrdr

=

=−⋅⋅ ⋅⋅ ⋅

∫

∫ (16)

The duration in time slots of a node in the SUCCESS

state is

(1) (1) (1) (1)

4

S rtsctsdataack

rts cts dataack

T tttt

t t tt

=+ ++++ ++

=+ +++ (17)

where rts

t, cts

t, data

t and ack

t are the duration times of

transmitting RTS, CTS, data and ACK frames,

respectively.

After the corresponding parts are replaced in Equation

(5), the throughput of RTS-DTDR is obtained. Following

the similar process, we can calculate the overhead of

RTS-DTDR from Equation (8).

4.4. Basic Directional Transmission and

Directional Reception (B-DTDR)

Since there is no RTS and CTS frames, B-DTDR has a

narrower bound on

C

T

(from 1

1

T

τ

=

to

2

data ack

T tt

=+ +

). Then we can calculate

C

T

by using

Equation (11).

And the success period time is

2

Sdata ack

T tt

=+ +

(18)

1

P

keeps the same as RTS-DTDR.

2

P

is equal to the

probability that no node transmits toward node

j

X

within

data

t

period and does not transmit in the slot when

node

i

X

begins to transmit with node

j

X

, therefore, we have

2 2

( 1)

2 2

2

data

p Stp S

P ee

θ θ

ρ ρ

π π

−+−

=⋅ (19)

Similarly, we have

3

4 4

( 1)

2

3

( 1)

2 2

4

ack

data

pS t

p Sp St

P e

P ee

θρ

π

θ θ

ρ ρ

π π

− +

− −+

=

= ⋅

(20)

Then after replacing the corresponding parts in

Equation (5), we get the throughput of B-DTDR. Since

2

ctrl ack

t t

= +

in B-DTDR, we can calculate the overhead

of B-DTDR from Equation (8).

4.5. Numerical Results

We compare the performance of the RTS-DTDR and

BDTDR under the different configurations and present

the results in Figure 6 and Figure 7.

Figure 6 shows the saturation throughput and

overhead of RTS-DTDR and B-DTDR under different

node density (

10,20,30,40

N

=

, respectively) when the

beamwidth is less than

6

π

. The results are obtained under

a short data length, i.e.,

40

data

t

τ

=. With the increased

node density, both the throughputs of RTS-DTDR and B-

DTDR begin to degrade although B-DTDR has a much

higher throughput than RTSDTDR protocol. The peak

value of B-DTDR is almost 20% higher than that of

RTS-DTDR. One possible reason is that when the

beamwidth is quite narrow, the number of the interfering

nodes is so small that those nodes cause nearly no

collisions. In this situation, RTS/CTS frames are not

necessary to be used. On the contrary, they only

contribute additional overhead on the throughput.

Then we calculate the throughput and overhead under

the long data length setting (i.e.,

120

data

t

τ

=and the

results are shown in Figure 7. Similarly, both RTS-

DTDR and B-DTDR perform well under a narrow beam

(e.g.,

15

π

). Under this setting, B-DTDR still has a higher

throughput than RTS-DTDR because it gets rid of the

bulky RTS/CTS mechanism. However, when the

beamwidth is increased further, the collisions caused by

292 H. N. DAI ET AL.

interfering nodes become remarkable, both the

throughput of RTS-DTDR and B-DTDR degrades.

5. Discussions

The results in Figure 6 and Figure 7 show that, when the

beamwidth is decreased, a higher network throughput can

be obtained. The capacity analysis in [15] also proves

that the capacity grows with the lessened beamwidth.

However, the capacity will not grow arbitrarily high

when the beamwidth decreases further and even

approaches to zero. Yi et al. [15] have also observed that

when the beamwidth is too small, the interference has

been fully reduced and there is no further improvement

by decreasing the beamwidth of the antennas.

Actually, when the beamwidth is narrow enough (more

specifically, less than a certain angle) a trans-mission can

yield a high success probability. As shown in Section 3.3,

if the beamwidth is less than

12

π

(i.e., 15°) and both

directional antennas are used at the transmitter and the

receiver, then the probability of a successful transmission

is greater than 99%. The transmission under this situation

can be regarded as a collision-tolerant transmission (the

collision probability is quite small). Hence, DTDR

should be the best way to use directional antennas.

Meanwhile, the angle 15° is feasible in most intelligent

directional antennas. Under this condition, the

complicated collision avoidance mechanisms, such as

RTS/CTS, are not necessary to be used because they only

contribute excessive overhead on the performance. At

that time, using some simple collision avoidance

mechanisms, such as the exponential back-off, might be

enough to reduce the interference.

(a) N=10

(b) N=20

(a) N=30

(b) N=40

Figure 6. Throughput comparison when

0.1,5 ,40

τ τ

== ===

rts cts ackdata

pt t tt(short data frame).

COLLISION-TOLERANT TRANSMISSION WITH NARROW-BEAM ANTENNAS 293

This collision-tolerant transmission gives us some

important implications on MAC design. Directional

antennas have different properties, e.g., higher spatial

reuse and the smaller interfering region. Although

RTS/CTS schemes work well in wireless networks using

omni-directional antennas, they cannot mitigate

interference caused by directional antennas completely

[5]. Thus, the MAC layer design with directional

antennas should start from another different perspective.

For example, when the beamwidth is narrow enough and

the collision probability is small, we can turn off

RTS/CTS. On the contrary, we should consider other

techniques, such as power control and multi-channel

schemes to reduce interference.

6. Conclusions

This paper studies the performance of wireless networks

using directional antennas with a narrow beam. In

particular, we examine the probability of a successful

transmission under Directional Transmission and

Directional Reception. The numerical results show that

the interference probability is quite low when the antenna

beamwidth is narrow enough. These results encourage us

to design a lightweight MAC protocol which turns off

RTS/CTS. The evaluation results prove that the protocol

has a higher throughput than the typical MAC protocol

based on RTS/CTS. The results also demonstrate that a

collision-tolerant transmission is feasible when the

beamwidth is narrow enough. One of our future works is

to implement the lightweight MAC protocol in simulators

and conduct experiments in real environments.

7. Acknowledgements

This research was partially supported by Natural Science

(a) N=10

(b) N=20

(a) N=30

(b) N=40

Figure 7. Throughput comparison when

0.1,5 ,120

τ τ

== ===

rts cts ackdata

pt t tt(long data frame).

294 H. N. DAI ET AL.

Foundation of China grant No.60573138 and 60773091,

and the National Grand Fundamental Research 973 Pro-

gram of China under Grant No.2006CB303000.

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