On local connectivity of cognitive radio ad hoc networks with directional antennas

Full text

On Local Connectivity of Cognitive Radio Ad Hoc
Networks with Directional Antennas
Yuanyuan Wang∗, Qiu Wang∗, Hong-Ning Dai∗, Haibo Wang†, Zibin Zheng‡and Jianqing Li∗
∗Macau University of Science and Technology, Macau SAR
wwwangyuanyuan@gmail.com; qiu wang@foxmail.com; hndai@ieee.org; jqli@must.edu.mo
†Beijing Jiaontong University, Beijing, P. R. China
hbwang@bjtu.edu.cn
‡Sun Yat-sen University, Guangzhou, P. R. China
zibin.gil@gmail.com
Abstract—In conventional cognitive radio ad hoc networks,
both primary users (PUs) and secondary users (SUs) are typically
equipped with omni-directional antennas, which can cause high
interference. As a result, SUs usually suffer from the poor
network connectivity due to the high interference caused by PUs.
Different from omni-directional antennas, directional antennas
can concentrate the transmission on some desired directions and
reduce interference on other undesired directions. In this paper,
we investigate the network connectivity of a novel cognitive radio
ad hoc network with directional antennas (DIR-CRAHNs), in
which both PUs and SUs are equipped with directional antennas.
We derive the local connectivity (i.e., probability of node isolation)
of such DIR-CRAHNs in closed form and numerical results show
that the proposed DIR-CRAHNs have higher local connectivity
than conventional cognitive radio ad hoc networks.
I. INTRO DUC TIO N
We have seen the growing demands for higher data rate due
to the emergence of bandwidth-ravenous applications (e.g., 4k
video streaming). The current wireless networks cannot offer
a solution to the drastic growth of high band-width demands
mainly owing to the underutilized radio spectrum. One reason
of the underutilized wireless spectrum is the fixed spectrum
allocation scheme, in which multiple channels are allocated
to avoid the interference. In this technology, radio spectrum is
allocated for legitimate users. Once the spectrum being allo-
cated for the dedicated users, no unsubscribed users can use
the spectrum. However, this fixed spectrum allocation scheme
is less efficient because a large portion of the licensed spectrum
has been underutilized especially when legitimate users are
idle. Recently, cognitive radio technology has been proposed
to improve the spectrum reusage by allowing unlicensed users
to use the spectrum opportunistically [1].
The network connectivity, as a fundamental property in
wireless ad hoc networks (WAHNs), has received extensive
attentions [2], [3]. The network connectivity measures the
probability that a sender can successfully communicate with
its receiver. Different from a homogeneous WAHN, a cognitive
radio wireless ad hoc network (CRAHN) consists of two types
of users: (i) Primary users using the licensed spectrum and
(ii) Cognitive (Secondary) users using the licensed spectrum
opportunistically (i.e., second users can only use the spectrum
when no primary users are using the spectrum). For simplicity,
we call the network formed by primary users as PNets and
call the network formed by Secondary users as SNets. The
network connectivity of SNets is usually more difficult to
be guaranteed than that of PNets due to the less available
spectrum for cognitive users than that for primary users [4],
[5], [6]. Thus, we concern about the network connectivity of
SNets in this paper. In such CRAHNs, both PNets and SNets
consist of users equipped with only omni-directional antennas,
which cause high interference. We name such CRAHNs with
omni-directional antennas as OMN-CRAHNs.
Recent works such as [7], [8], [9] found that applying
directional antennas instead of omni-directional antennas to
wireless networks can greatly improve the network perfor-
mance (e.g., capacity and connectivity). Compared with omni-
directional antennas, directional antennas can concentrate the
radio signals on desired directions. In other undesired direc-
tions, there are no radio signals or weakened signals. Thus,
using directional antennas in wireless networks can potentially
reduce the interference and improve the spectrum reuse, and
consequently improve the network performance. One of in-
teresting questions is whether using directional antennas in
CRAHNs can also improve the network performance.
In this paper, we formally propose cognitive radio ad hoc
networks equipped with directional antennas named as DIR-
CRAHNs that have the following characteristics.
•Each primary user (node) is mounted with a directional
antenna, which can legitimately use the licensed spec-
trum.
•Each secondary user (node) is mounted with a directional
antenna, which can use the licensed spectrum opportunis-
tically only when no PUs are using the spectrum.
•The primary nodes form an ad hoc network (i.e. PNet)
and the secondary nodes form an ad hoc network (i.e.,
SNet).
There are few studies on DIR-CRAHNs. Though Wei et
al. investigated the asymptotic throughput of CRAHNs with
directional transmissions in [10], they derived the network
throughput based on the asymptotically connected CRAHNs
with simple extension from OMN-CRAHNs. Their study does
not offer a general analytical framework on the local network

connectivity for DIR-CRAHNs. To the best of our knowledge,
there is no study on establishing the analytical model on the
local network connectivity of DIR-CRAHNs. To investigate
the local network connectivity of DIR-CRAHNs is important
because (i) the full network connectivity (i.e., the probability
that each node can connect to any other node in the network)
of SNet is usually more difficult to be ensured than the
local network connectivity since the secondary users have less
spectrum to use and (ii) the derivation of the local network
connectivity is the prerequisite of obtaining the overall net-
work connectivity of ad hoc networks [2]. Therefore, the goal
of this paper is to investigate the local network connectivity
of DIR-CRAHNs.
In this paper, we investigate the local network connectivity
of DIR-CRAHNs, which is believed to be one of first studies
in this new area. The contributions of this work can be
summarized as follows.
•We formally identify DIR-CRAHNs that characterize
the features of cognitive radio ad hoc networks with
directional antennas.
•We propose a theoretical framework to investigate the
local network connectivity of such DIR-CRAHNs.
•Our analytic model is quite general since we consider
both DIR-CRAHNs and OMN-CRAHNs in the same
theoretical framework and OMN-CRAHNs may become
a special case of DIR-CRAHNs in some scenarios.
•Our results show that DIR-CRAHNs can provide a better
network connectivity than conventional OMN-CRAHNs
due to the usage of directional antennas, which signifi-
cantly reduce the interference and improve the spectrum
reuse.
The rest of this paper is organized as follows. In Section II,
we present the models used in this paper. Section III analyzes
the local connectivity of DIR-CRAHNs and OMN-CRAHNs.
In Section IV, we present the numerical results on the local
network connectivity. Finally, Section V concludes this paper.
II. SY S TE M MO DE L S
Section II-A presents the network model. Antenna models
will be introduced in Section II-B. Section II-C then presents
the channel model. We next define link criterion for SUs in
Section II-D.
A. Network Model
There are several cognitive radio spectrum sharing
paradigms in CRAHNs: underlay paradigm, overlay paradigm
and interweave paradigm [11]. In this paper, we only consider
the overlay paradigm because it fully utilizes the spectrum [?].
In CRAHNs, there are two kinds of users: Primary users (PUs)
and Secondary users (SUs). In the over-laid paradigm, PUs
have the first priority to use the licensed spectrum and SUs
can only use the spectrum opportunistically (e.g., when no PUs
are using the spectrum). For simplicity, we call the network
formed by PUs as PNets and call the network formed by SUs
as SNets. In this paper, we mainly concentrate on the network
connectivity of SNets since the network connectivity of PNets
PU
SU
Communication link
Unavailable link
Interfered SU
PU1
PU2
SU1
SU2
PU3
SU3
Fig. 1: Network model of CRAHNs
is usually ensured due to the higher spectrum availability of
PUs than that of SUs.
We assume that both PUs and SUs are distributed according
to a two-dimensional homogenous Poisson point process with
density λpand λs, respectively. Without loss of generality,
we assume that all PUs and SUs use the same transmission
power. Besides, the traffic arrival rate of PUs follows a Poisson
process with density λarr. Note that PUs have the higher
priority to use the spectrum and we assume that every PU
can access all the spectrum all the time. In this paper, we
focus on the impacts of PUs on the network connectivity of
SNets. To protect the connectivity of PNets, whenever there is
an active PU within the transmission range of an SU, this SU
will be prohibited for transmission. That is the reason why the
network connectivity of SNets is usually more difficult to be
ensured than that of PNets [4], [5]. Take Fig. 1 as an example
of CRAHNs, where red ovals represent PUs and blue hexagons
represent SUs. In particular, PU1can establish a link to PU2
and PU2can establish a link to PU3(i.e., the solid lines in
Fig. 1) while SU1cannot establish the link to SU2(i.e., the
dash lines). The reason lies that PUs have the higher priority
to use the spectrum than SUs and both SU1and SU2within
the transmission regions of PU1and PU2have to be silent.
B. Antenna Model
In conventional CRAHNs, both PUs and SUs are typically
equipped with omni-directional antennas, which radiate/collect
radio signals into/from all directions equally, as shown in
Fig. 2. In our proposed DIR-CRAHNs, both PUs and SUs
are equipped with directional antennas only, which can con-
centrate transmitting or receiving capability on some desired
directions. As a result, the received/emitted signals by direc-
tional antennas can be significantly enhanced compared with
omni-directional antennas.
We often use the antenna gain to model the transmitting
or receiving capability of an antenna. It is complicated to
compute the antenna gain of a realistic antenna in each
direction. Besides, realistic antenna model can not be used
to solve the problem of deriving the optimal bounds on the
network connectivity [12]. Thus, an approximated antenna
model has been proposed in [13]. This model is named as
Sector antenna model. Sector model only consists of one main
beam with beamwidth θmand all the side/back lobes are
ignored, as shown in Fig. 3. Following the analysis in the
previous study [9], we have the antenna gain Gdof Sector

Fig. 2: Omni-directional antenna
m
q
Fig. 3: Sector antenna
model as follows,
Gd=2
1−cos θm
2
.(1)
As shown in Eq. (1), Gdis a function of the beamwidth
θm. Obviously, when θm= 2π, we have the gain of an omni-
directional antenna Go= 1.
C. Channel Model
We assume that a node transmits with power Pt. Then,
the receiving power at the receiver denoted by Prcan be
calculated by
Pr=Pt·Gt·Gr
10ω/10 ·lα,(2)
where lis the distance between the transmitter and the receiver,
Gtand Grare the antenna gains of a transmitter and a
receiver, respectively. In outdoor environment, the channel
gain between a transmitter and a receiver is mainly determined
by the path loss effect and the shadow fading effect. In
particular, the path loss effect is characterized by the path
loss exponent α(usually 2≤α≤6[14]). The shadow fading
effect is modelled as log-normal random variable ωwith zero
mean and standard deviation σ(dB).
In practice, we usually compute the power attenuation [3]
between two nodes instead of computing the received power
Pr. We then introduce the power attenuation ∆defined as
follows,
∆ = Pt
Pr
=10ω/10 ·lα
Gt·Gr
.(3)
D. Link Criterion
It is more challenging to analyze the link condition (cri-
terion) in CRAHNs than that in conventional WAHNs. In
conventional WAHNs, two nodes can establish a link between
them if they fall into the transmission region of each other.
However, in CRAHNs, the link criterion of two SUs depends
on not only the transmission region but also the spectrum
availability.
In this paper, we consider the bidirectional link since it
can guarantee the deliver of acknowledgement in practical
networks (e.g., ACK has been used in most of Carrier sense
multiple access (CSMA) media access control (MAC) proto-
cols). The bidirectional link means that node SUiand SUjin
can establish a link if and only if SUiand SUjcan transmit
successfully to each other. Then, we define the link condition
of DIR-CRAHNs and the link condition of OMN-CRAHNs,
respectively.
Definition 1: Link Condition of DIR-CRAHNs. In DIR-
CRAHNs, two nodes SUiand SUjare connected if and only
if both the following conditions are satisfied.
1) The Euclidean distance between SUiand SUjis less
than rd, i.e., kSUi−SUjk< rd, where SUiand SUj
denote the positions of two SUs in a 2-D plane, rdis
the transmission range of an SU and k · k denotes the
Euclidean distance;
2) both SUiand SUjhave the available channel, which
means that there are no active PUs in SUs transmission
region or the PUs shall not point their antennas to SUs
if they fall into SUs transmission region.
Note that in DIR-CRAHNs, we assume that the main
directions of each pair of transmitter and receiver are pointed
to each other to ensure the best link quality by using beam-
locking mechanisms [15]. Besides, each SU in our DIR-
CRAHNs is equipped with a single directional antenna, which
can arbitrarily switch its antenna direction forward or back-
ward. It can be easily achieved by Switched Beam antenna
[16] or Adaptive Array antenna [17].
We then give the link condition of OMN-CRAHNs as
follows.
Definition 2: Link Condition of OMN-CRAHNs. In OMN-
CRAHNs, two nodes SUiand SUjcan establish a link if and
only if both the following conditions are satisfied.
1) The Euclidean distance between SUiand SUjis less than
ro, i.e., kSUi−SUjk< ro, where rois the transmission
range of SUs;
2) both SUiand SUjhave the available channel, which
means that there are no active PUs in the transmission
region of both SUiand SUj.
The transmission range rocan be obtained by Eq. (3). In
particular, we let the power attenuation fixed at a given value
∆t(a threshold) and Gt=Gr= 1, and then substitute them
in Eq. (3). We next have
ro=∆t
10ω/10 1
α
.(4)
The transmission range rdof DIR-CRAHNs can be calcu-
lated in a way similar to OMN-CRAHNs. Specifically, We
also let Eq. (3) be equal to the given threshold ∆tand
Gt=Gr=Gdthen we have
rd=∆t·G2
d
10ω/10 
1
α
.(5)
We assume that both OMN-CRAHNs and DIR-CRAHNs
have the same network topologies and the environment settings
(e.g., the identical shadowing effect factor ωand σ). We then
obtain the relationship between rdand roby combining Eq.
(4) and Eq. (5), which is given by
rd=roG
2
α
d.(6)
In our DIR-CRAHNs, we assume PUs and SUs have the
different directional antenna settings. In particular, each SU
is equipped with an antenna with beamwidth θsand each PU
is equipped with an antenna with beamwidth θp. Then, the
transmission region rdcan be expressed as
rd=ro 2
1−cos θs
2!2
α
.(7)

It is implied from Eq. (7) that DIR-CRAHNs can extend the
transmission range of SUs by 2
1−cos θs
2
2
α, compared with
OMN-CRAHNs.
III. LO CA L NE TW ORK CO NNE CT I VI TY
In this section, we analyze the local connectivity of both
DIR-CRAHNs and OMN-CRAHNs. We first derive the suc-
cessful transmission probability of SUs in Section III-A, which
will then be used to derive the probability of node isolation
in Section III-B.
A. The Successful Transmission Probability of SUs
We first analyze the successful transmission probability of
SUs denoted by Pst
SU . The successful transmission of an SU
requires no active PUs in its transmission region. Hence, we
need to consider the activities of PUs. In particular, we have
the probability of qpackets arriving at a PU in unit time
is Parr
P U (q) = λq
arr
q!e−λarr since the traffic arrival event at a
PU follows a Poisson process with the density λarr as given
in Section II-A. Thus, the probability that there is no traffic
arriving at a PU in unit time is Parr
P U (0) = e−λarr .
The probability that there are nPUs falling in the transmis-
sion region of an SU is PP U (n) = (Asλp)n
n!e−Asλp, where As
is the area of transmission region of an SU in DIR-CRAHNs
according to the following equation,
As=θsr2
d
2=θsr2
o
2 2
1−cos θs
2!4
α
.(8)
Note that Eq. (8) can also represent the transmission area
of an SU in OMN-CRAHNs, i.e., when θs= 2π,As=πr2
o.
We next derive Pst
SU , which is essentially equal to the
probability that there is no traffic arriving at nPUs in the
transmission region of an SU. Importantly, since there is a
uniform distribution of antenna directions of PUs and PUs are
distributed according to a homogeneous Poisson process, there
are only θp
2π·nPUs can interfere with this SU. Therefore, we
have Pst
SU as follows,
Pst
SU =
∞
X
n=0
PP U (n)·Parr
P U (0)n·θp
2π
=
∞
X
n=0
(Asλp)n
n!e−Asλp·e−λarr nθp
2π
= exp n− Asλp1−e−λarrθp
2πo
= exp n−(2
1−cos θs
2
)4
αθsr2
o
2λp1−e−λarr θp
2πo.
(9)
It is shown in Eq. (9) that the SU successful transmission
probability Pst
SU is dependent on λp,λarr,θsand θpin a DIR-
CRAHN. Besides, Pst
SU is decreasing with As,λp,λarr and
θp. When θs=θp= 2π, our analysis becomes the case of an
OMN-CRAHN.
P
q
PU
S
q
d
r
j
SU
i
SU
(a) Two nodes of DIR-
CRAHNs network
o
q
i
SU
o
r
l
j
SU
PU
(b) Two nodes of OMN-
CRAHNs network
Fig. 4: Local connectivity analysis of two adjacent nodes with
different antenna models
B. Probability of Node Isolation
We concern with the probability of node isolation, denoted
by Piso, as the metric to measure the local network connec-
tivity [2], [8]. The probability of node isolation is defined
as the probability that a node can not connect to any other
nodes in the network. According to the link condition of both
OMN-CRAHNs and DIR-CRAHNs in Section II-D, we give
a general formula of Piso in CRAHNs as follows,
Piso =∞
X
n=0
PSU (n)·1−Pst
SUi+Pst
SUi·1−Pst
SUj|SUin,
(10)
where PSU (n)is the probability that there are nSUs falling
in the transmission region of an SUi,Pst
SUiis the probability
that an SUican successfully transmit and Pst
SUj|SUiis the
probability that an neighboring SUjcan successfully transmit
under the condition that SUisuccessfully transmits. It is
obvious that PS U (n) = (Asλs)n
n!e−Asλsand Pst
SUi=Pst
SU
(given by Eq. (9)).
We next analyze the condition probability Pst
SUj|SUi. We cat-
egorize the derivations of Pst
SUj|SUiaccording to two different
cases for DIR-CRAHNs and OMN-CRAHNs as shown in Fig.
4(a) and Fig. 4(b), respectively.
Case I: DIR-CRAHNs
Fig. 4(a) shows the scenario of any two adjacent nodes in a
DIR-CRAHN. According to link condition of DIR-CRAHNs
(as shown in Section II-D), we know that if there are no
active PUs directing their antennas towards SUior SUjif
they are falling in the transmission region of SUior SUj, then
SUior SUjcan transmit successfully. We then conclude that
the probability that SUican transmit successfully denoted by
Pst
SUiis independent with the probability that SUjcan transmit
backward successfully denoted by Pst
SUj. Therefore, we have
Pst
SUj|SUiin DIR-CRAHNs as follows,
Pst
SUj|S Ui=Pst
SUj=Pst
SU .(11)
We substitute Eq. (11) into Eq. (10), combine Eq. (9), and
then obtain Piso in DIR-CRAHNs as follows,
Piso = 1 −exp −Asλp·1−e−λarr
θp
2π+
exp 


As
−λp1−e−λarr
θp
2π−λse
−Asλp 1−eλarr
θp
2π!



,
(12)

where As=θsr2
o
2(2
1−cos θs
2
)4
α, which is given by Eq. (8).
Case II: OMN-CRAHNs
Fig. 4(b) shows the scenario of any two adjacent nodes in an
OMN-CRAHN. If SUican transmit successfully, there must
be no active PUs falling in the transmission region of SUi.
Different from Case I, there must be no active PUs falling
in the intersection of the transmission regions of both SUi
and SUjsince an omni-directional antenna can receive signals
from all directions. Therefore, the successful transmission of
SUjonly requires the condition that no active PUs fall in the
shaded region as shown in Fig. 4(b). We then derive the shaded
area as follows,
SO(l) = (π−θo)r2
o+lrosin θo
2,(13)
where θo= 2 arccos l
2ro, and lis the Euclidean distance
between SUiand SUj, which is a random variable. We then
have the expectation of SOas follows,
E[SO(l)] = Zro
0
SO(l)·fl(l)dl, (14)
where fl(l)is the probability density function (PDF) of l.
We next calculate fl(l). Since SUs are randomly distributed
in a 2-D area according to homogeneous Poisson point pro-
cess, we then have the PDF of las follows,
fl(l) = 2πl
πr2
o
=2l
r2
o
.(15)
We then can have E[SO(l)] = 3√3
4r2
oafter substituting Eq.
(15) and Eq. (13) into Eq. (14).
With the above analysis, the probability that SUjcan
successfully transmit after SUitransmits Pst
SUj|SUiin OMN-
CRAHNs is given by
Pst
SUj|S Ui=e−λpE[So(l)](1−e−λarr )=e−3√3
4r2
oλp(1−e−λarr ).
(16)
Finally, we can obtain the Piso in OMN-CRAHNs by
substituting Eq. (16) into Eq. (10), which is expressed as
Piso = 1 −exp n−λpπr2
o·(1 −e−λarr )o+
exp πr2
o−λp1−e−λarr −λse−3√3
4r2
oλp(1−e−λarr ).
(17)
IV. NUM E RI CA L RES ULTS
In this section, we present several sets of numerical results
to evaluate the local connectivity of DIR-CRAHNs in terms
of Piso. We first compare the results between DIR-CRAHNs
and OMN-CRAHNs with different settings including density
of PUs λp, density of SUs λsand path loss exponent α. Fig.
5 presents the comparative results. In particular, it is shown
in Fig. 5 that the probability of node isolation Piso increases
with the increased node density of PUs λp, implying that
the connectivity of SNets is decreased. This is mainly due
to the increased interference caused by the increased number
of PUs. Besides, Fig. 5 also shows that Piso decreases with the
increased density of SUs λs, which implies that the network
connectivity of SNets is improved (if we compare Fig. 5 (a),
Fig. 5 (b), Fig. 5 (c) and Fig. 5 (d) together). Moreover, we
can see in Fig. 5 that DIR-CRAHNs always have the lower
probability of node isolation Piso than that of OMN-CRAHNs,
indicating that DIR-CRAHNs provide better connectivity than
OMN-CRAHNs under different density of PUs λp, density
of SUs λsand path loss exponent α. The improvement of
network connectivity of DIR-CRAHNs over OMN-CRAHNs
main owes to the reduced interference from PUs to SUs by
using directional antennas.
We then evaluate the network connectivity of DIR-CRAHNs
with different system parameters. In particular, Fig. 6 and
Fig. 7 show the probability of node isolation Piso of DIR-
CRAHNs with different values of path loss exponent α= 3
and α= 5, respectively. First, both Fig. 6 and Fig. 7 show the
similar trends to Fig. 5, i.e., Piso increases with the increased
node density of PUs and decreases with the increased node
density of SUs, implying that the network connectivity of
SNets heavily depends on PUs’ activities. Besides, it is shown
in Fig. 6 and Fig. 7 that the probability of node isolation Piso
increases when the antenna beamwidth of PUs θpis increased.
For example, comparing Fig. 6 (a) with Fig. 6 (b), we can find
that Piso increases when the antenna beamwidth θpis increased
from 30◦to 45◦while θsis fixed. This is because more SUs
are suffering from the outage due to the increased interference
region of PUs when the antenna beamwidth of PUs θpis
increased. On the contrary, increasing the antenna beamwidth
of SUs θscan decrease Piso , implying the improved network
connectivity. Take Fig. 6 (a) and Fig. 6 (c) as examples
again. We find that Piso significantly decreases when the
antenna beamwidth θsis increased from 30◦to 45◦while
θpis fixed to be 30◦. The network connectivity improvement
may owe to the expanded transmission region of SUs, which
results in the increased number of neighbors. Furthermore, the
local connectivity also depends on the channel condition. For
example, increasing the path loss exponent (e.g., αis increase
from 3 to 5) can lead to the decrement of Piso when we
compare Fig. 6 with Fig. 7.
V. CO NCL US I ON
In this paper, we propose cognitive radio ad hoc networks
equipped with directional antennas (DIR-CRAHNs), and for-
mally establish an analytical framework to investigate the local
connectivity of DIR-CRAHNs. Compared with conventional
cognitive radio ad hoc network, our DIR-CRAHNs can im-
prove the network connectivity. The connectivity improvement
mainly owes to (i) the reduced PU-to-SU interference by
adopting directional antenna in PUs and (ii) the extended
transmission range of SUs by using directional antennas.
ACK NOWL E DG E ME NT
The work described in this paper was partially supported
by Macao Science and Technology Development Fund under
Grant No. 096/2013/A3 and partially supported by the Na-
tional Natural Science Foundation of China under Project No.
61471026. The authors would like to thank Gordon K.-T. Hon
for his constructive comments.

0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
DIR−CRAHNs
OMN−CRAHNs
(a) λs= 0.1,α= 3,
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
DIR−CRAHNs
OMN−CRAHNs
(b) λs= 1,α= 3
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
DIR−CRAHNs
OMN−CRAHNs
(c) λs= 0.1,α= 5
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
DIR−CRAHNs
OMN−CRAHNs
(d) λs= 1,α= 5
Fig. 5: Probability of node isolation Piso of DIR-CRAHNs versus OMN-CRAHNs. System parameters are given as ro= 1,
λarr = 0.3,θp= 30◦,θs= 60◦.
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
λs=0.1
λs=1
(a) θp= 30◦,θs= 30◦
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
λs=0.1
λs=1
(b) θp= 45◦,θs= 30◦
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
λs=0.1
λs=1
(c) θp= 30◦,θs= 45◦
Fig. 6: Probability of node isolation Piso of DIR-CRAHNs. System parameters are given as ro= 1,λarr = 0.3,α= 3.
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
λs=0.1
λs=1
(a) θp= 30◦,θs= 30◦
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
λs=0.1
λs=1
(b) θp= 45◦,θs= 30◦
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
The density of PUs λp
Piso
λs=0.1
λs=1
(c) θp= 30◦,θs= 45◦
Fig. 7: Probability of node isolation Piso of DIR-CRAHNs. System parameters are given as ro= 1,λarr = 0.3,α= 5.
REF ERE NC E S
[1] I. Akyildiz, W.-Y. Lee, and K. Chowdhury, “CRAHNs: Cognitive radio
ad hoc network,” Ad Hoc Network, vol. 7, no. 5, pp. 810 – 836, 2009.
[2] C. Bettstetter, “On the Connectivity of Ad Hoc Networks,” The Com-
puter Journal, vol. 47, no. 4, pp. 432 – 447, 2004.
[3] C. Bettstetter, C. Hartmann, and C. Moser, “How does randomized
beamforming improve the connectivity of ad hoc networks,” in Pro-
ceedings of IEEE ICC, 2005.
[4] W. C. Ao, S.-M. Cheng, and K.-C. Chen, “Connectivity of multiple
cooperative cognitive radio ad hoc networks,” Selected Areas in Com-
munications, IEEE Journal on, vol. 30, no. 2, pp. 263–270, February
2012.
[5] J. Liu, Q. Zhang, Y. Zhang, Z. Wei, and S. Ma, “Connectivity of two
nodes in cognitive radio ad hoc networks,” in Proc. IEEE WCNC, 2013.
[6] D. Zhai, M. Sheng, X. Wang, and Y. Zhang, “Local Connectivity of
Cognitive Radio Ad Hoc Networks,” in IEEE GLOBECOM, 2014.
[7] H.-N. Dai, K.-W. Ng, R. C.-W. Wong, and M.-Y. Wu, “On the Capacity
of Multi-Channel Wireless Networks Using Directional Antennas,” in
Proceedings of IEEE INFOCOM, 2008.
[8] X. Zhou, S. Durrani, and H. Jones, “Connectivity Analysis of Wirelss
Ad Hoc Networks with Beamforming,” IEEE Transactions on Vehicular
Technology, vol. 58, no. 9, pp. 5247 – 5257, 2009.
[9] Y. Wang, H.-N. Dai, Q. Wang, X. Li, Q. Zhao, and C. F. Cheang, “Local
Connectivity of Wireless Networks with Directional Antennas,” in IEEE
PIMRC, 2015.
[10] Z. Wei, Z. Feng, Q. Zhang, W. Li, and T. Gulliver, “The asymptotic
throughput and connectivity of cognitive radio networks with directional
transmission,” Communications and Networks, Journal of, vol. 16, no. 2,
pp. 227–237, April 2014.
[11] A. Goldsmith, S. Jafar, I. Maric, and S. Srinivasa, “Breaking spectrum
gridlock with cognitive radios: An information theoretic perspective,”
Proceedings of the IEEE, vol. 97, no. 5, pp. 894–914, May 2009.
[12] M. Kiese, C. Hartmann, and R. Vilzmann, “Optimality bounds of the
connectivity of adhoc networks with beamforming antennas,” in IEEE
GLOBECOM, 2009.
[13] R. Ramanathan, “On the performance of ad hoc networks with beam-
forming antennas,” in Proceedings of ACM MobiHoc, 2001.
[14] T. S. Rappaport, Wireless communications : principles and practice,
2nd ed. Upper Saddle River, N.J.: Prentice Hall PTR, 2002.
[15] M. Takai, J. Martin, R. Bagrodia, and A. Ren, “Directional virtual
carrier sensing for directional antennas in mobile ad hoc networks,”
in Proceedings of ACM MobiHoc, 2002.
[16] T. Korakis, G. Jakllari, and L. Tassiulas, “A MAC protocol for full
exploitation of directional antennas in ad-hoc wireless networks,” in
Proceedings of ACM MobiHoc, 2003.
[17] R. Ramanathan, J. Redi, C. Santivanez, D. Wiggins, and S. Polit, “Ad
hoc networking with directional antennas: A complete system solution,”
IEEE JSAC, vol. 23, no. 3, pp. 496–506, 2005.