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Opportunistic Spectrum Access in Cognitive Radio
Networks
Senhua Huang, Xin Liu, and Zhi Ding
University of California, Davis
Davis, CA 95616, USA
Abstract—Enabled by regulatory initiatives and radio technol-
ogy advances, opportunistic spectrum access has the potential to
mitigate spectrum scarcity and satisfy the increasing demand for
spectrum. In this paper, we consider a scenario where secondary
users can opportunistically access unused spectrum vacated by
idle primaries. We introduce two metrics to protect primary
performance, namely collision probability and overlapping time.
We present two spectrum access schemes using different sensing,
back-off, and transmission mechanisms. We show that they
achieve indistinguishable secondary performance under given
primary constraints. We provide closed form analysis on sec-
ondary user performance, present a tight capacity upper bound,
reveal the impact of various design options, such as sensing,
packet length distribution, back-off time, packet overhead, and
grouping. Our work sheds light on the fundamental properties
and design criteria on opportunistic spectrum access.
I. INTRODUCTION
The breakneck proliferation of wireless devices and rapid
growth of wireless services continue to stretch the limited
spectral resource. In fact, most spectrum bands suitable for
terrestrial wireless communication have already been allocated
by the regulatory agencies to existing licensees. On the other
hand, the current approach of static spectral allocation is
highly wasteful. Measurement has shown that over 60% of
the licensed spectrum below 6GHz remains unused or under-
utilized [1], [2]. To exploit the reported “white space” in exist-
ing bands [1], cognitive radio networks have been considered
as the viable technology to improve spectral efficiency. With
primary licensee’s consent, secondary users equipped with
cognitive radios may be allowed to transmit on primary bands
when the Primary users are inactive [3], [4], [5].
Serious challenges must be resolved in order for the cogni-
tive radios to be acceptable. First, secondary users must not
be disruptive to primary user communications. Secondly, an
access mechanism is required to reduce contention between
secondary users to efficiently share spectrum opportunities.
However, coordination and synchronization among secondary
users may be limited due to the decentralized nature of sec-
ondary user access, particularly if secondary users of different
networks coexist. In addition, each secondary users may not
be able to sense all channels due to the limitation on hardware
or/and sensing capability, and thus algorithm to decide which
channel to monitor is needed.
In this paper, we focus on non-intrusive spectrum access
schemes that do not require primary users to alter theirexisting
hardware or behavior. The activities of primary users on
different frequency bands are modeled as independent M/G/1
queues, where secondary users can exploit a primary channel
when it is idle. We introduce two protection metrics (collision
probability and overlapping time), to which secondary users’
access schemes must abide in order to protect the quality
of service (QoS) of primary users. Such constraints can be
imposed by primary licensees or spectral regulators.
Under the constraints, we investigate the capacity of sec-
ondary users under various sensing-based random access
schemes. We propose two random access schemes for sec-
ondary users, namely, VX and KS schemes, with different
sensing, transmission, and back-off mechanisms. We study
extensively the throughput performance of secondary users
with the VX scheme. We thoroughly investigate both the
simple system setup with with one primary band and one
secondary user, and the more general case in which there
are multiple primary bands and multiple secondary users. Our
results illustrate the fundamental properties and design criteria
of opportunistic spectral access for cognitive radio networks.
The rest of the paper is organized as follows. We first
describe a number of related works in the field in Section II
before presenting our system model, performance metrics and
problem formulation in Section III. We present two random
access schemes, and analyze the throughput performance and
optimum access parameters in Section IV. Simulation results
are also given in Section IV to verify our analysis. We extend
the results to multi-band competitive systems in Section V.
We conclude the paper and discuss future work in Section VI.
II. RELATED WORK
To facilitate spectrum sharing, researchers have considered
the design of a common control channel to exchangespectrum
access and sensing information and facilitate collaborative
sensing and spectrum reservation/sharing, e.g., in [6], [7], [8].
Centralized and decentralized spectrum auction and brokerage
have been proposed for efficient spectrum sharing, e.g., in [9],
[10], [11]. Co-existence of cognitive users in unlicensed band
has also been studied [12], [13], [14].
Researchers have also considered sensing-based decentral-
ized cognitive medium access schemes [15], [16], [17]. In
[15], the authors model the states of primary bands as two-
state Markovian process and maximize the transmission rate
of secondary users in certain time slots. In [16], the authors
design a CSMA/CA-based cognitive radio MAC protocol that
uses channel statistics to determine the optimal access range
2
and the number of channels to access. In [17], the authors
develop a slotted transmission scheme of secondary user via
periodic channel sensing based on Constraint Markov Decision
Processes. In comparison, our model is more general. We do
not assume exponential busy periodof primary users (required
in Markovian models), neither synchronization between mul-
tiple users or feedback from receivers. Our work introduces
explicit guarantee on the performance of primary users and we
provide closed form analysis on the capacity limit of secondary
users under the primary constraints.
III. SYSTEM MODEL
Consider a system with Nhomogeneous primary bands
(channels) and Msecondary users (SUs). Primary users (PUs)
are the legacy users of these bands and thus have higher
priority over SUs. A primary band is called busy if it is used
by one or more PUs, and idle otherwise.
We assume the arrival process of a PU is Poisson while
the service time distribution can be arbitrary. This assumption
holds in many situations. For example, in a data network,
packet arrival process is Poisson while packet length distri-
bution can be arbitrary. Similarly, the arrival process of a data
session is often Poisson while the length of the data file can be
heavy tail. In addition, the call arrival process for voice traffic
can also be modeled as a Poisson process. When there are
multiple PUs in a band, the busy and idle periods of the band
can be modeled as the busy and idle periodsof an M/G/1 queue
(with multiple inputs). In this case, the idle time distribution is
exponential while the busy time distribution is general. From
the viewpoint of an SU, because the objective is to utilize
the idle period, one can treat the aggregated primaries as one
PU. Therefore, we assume, without loss of generality, there
is one PU per band and the activities in different bands are
independent and identically distributed (i.i.d.). We also assume
that the system is stationary and ergodic.
Let V1and L1be random variables denoting the idle period
and busy period of a primary band, respectively. Let v1=
E[V1]and l1=E[L1]. Let
α=v1
v1+l1
.
Note that αis the probability (or the percentage of time) that
a primary band is idle. In other words, a primary band is idle
αfraction of time, which could be exploited by SUs.
The focus here is on the non-intrusive spectrum access
schemes of cognitive radio devices. The non-intrusiveness has
two-folded meanings. On one hand, PUs are not required to
change their existing transmission strategies and algorithms to
coordinate with SUs. On the other hand, the access activities
of SUs should guarantee that the impact incurred on the
transmissions of PUs is insignificant. We also assume that
the PU’s channel access is not affected by SU’s behavior.
For example, PUs does not sense the channel before its
transmission. If a PU and a SU transmit simultaneously, the
PU does not retransmit or back-off. In other words, the idle
and busy periods of a primary band are not affected by SUs.
An SU performs sensing before transmission and only trans-
mits if the primary band is idle. We assume that SUs perform
perfect sensing, i.e., the false alarm and missing probability of
the sensing is zero. Additionally, we assume that the sensing of
the channel takes an infinitely small amount of time to finish.
Furthermore, due to the limit of radio front-end equipment,
we assume that the SU cannot sense the channel when it
is transmitting. We do not assume synchronization between
primary and secondary users or control channel among SUs.
A. Primary Protection
In our system, collision only happens in the following
scenario: a secondary user senses the channel idle and starts
transmission, and the primary user returns to the band before
the SU finishes its transmission. To capture the impact of
collision on PUs and thus to protect PUs, we introduce the
following metrics. The first constraint metric is the probability
of the collision observed by the PU. As shown in Fig. 1,
collision is defined as the event that the PU starts transmission
when the SU is transmitting a packet on the channel. Let Pc
1
and Pc
2be the collision probabilities observed by a PU and
an SU, respectively. Under the assumption of stationary and
ergodicity, we have:
Pc
1= lim
T−>∞
No. of collisions in [0,T]
No. of busy periods of PU in [0,T]
Pc
2= lim
T−>∞
No. of collisions in [0,T]
No. of packets transmitted for SU in [0,T] .
(1)
This constraint is suitable for the situation where the average
packet length of SUs, is close to that of PUs.
V2
2nd user
1
st user
V1L1
V2
scheme1:Vx Virtual
Xmit L2V2L2
Time
Tim
e
V2
Overlapping Time
R1 Sensing Point of SU
Successful Transmission
Collision!
2nd user
s
cheme2:Ks Tim
e
Successful Transmission
L2
Collision!
Fig. 1. Random Access Schemes of A Cognitive Radio User.
On the other hand, if a busy period of a PU consists of
one or more service sessions (e.g., the whole duration of a
telephone call or the total time to transfer a large file), then
only a very small portion of the primary’s traffic session is
affected by collision. For example, suppose that the PU is
having voice communication, the busy period (call session)
may be thousands times longer than the duration of a IP packet
transmitted by SUs. In this case, from the perspective of the
PU, the collision probability metric is not appropriate, and the
duration of the interruption caused by SUs is more important.
Therefore, the second constraint metric is the percentage of
3
overlapping time. One example of the overlapping time is
illustrated in Fig. 1 with length R1. Specifically, we have
the following definition of the percentage of overlapping time
observed by the PU:
Pr
1= lim
T−>∞
Length of overlapping time in [0,T]
T.(2)
To protect the transmission of PUs, the system can set the
following constraints:
Pc
1≤η
or, Pr
1≤r0,(3)
where ηand r0are performance thresholds predetermined by
the network operator of the primary bands and/or the spectrum
regulators. In fact, we will show that the above two constraints
are closely related in the following sections.
B. Objective Function
We assume that there are always packets waiting for trans-
mission for the SU. So the results obtained can be regarded as
an upper bound on the capacity of SUs. We also assume that
SUs have the knowledge of the statistics of the channels. In
particular, each SU knows the average of busy period l1and
idle period v1, i.e., the average channel occupancy behavior
of the PUs. This knowledge can be obtained by the SU from
historic information, measurement results, and/or database.
SUs also have the information of the access constraints posed
by the PUs or government regulators, i.e., ηor r0, which are
predefined when they are admitted into the network.
The objective of the spectrum access is to maximize the
achievable capacity (throughput) of the SU, denoted by C2.
More specifically, C2is defined as the time proportion that an
SU transmits on the channel without collision. Given that α
is the fraction of time the PU is idle, we have C2≤α. The
resulting optimization problem is formulated as below:
max{C2}
s.t.
Pc
1≤η
or, Pr
1≤r0.
(4)
In other words, an SU can decide its sensing scheme, access
scheme, packet length and distribution, and back-off duration
and distribution to maximize its capacity. For the rest of the
paper, let L2be a random variable denoting the secondary
transmission time, which is refereed to as packet length. Let
V2be a random variable denoting the back-off time, which is
also referred to as vacation. Let l2=E[L2]and v2=E[V2].
IV. ONE PRIMARY BAND,ONE SU
In this section, we first consider the case where there is only
one SU and one primary band. We analyze the throughput
performance of the SU under the constraints posed by the PU.
A. Random Access Schemes
The media access schemes (or protocol) we consider in this
paper are illustrated in Fig. 1 and described as below:
•VX Scheme (Virtual-Xmit-if-Busy): The SU senses the
channel. If the channel is idle, the SU transmits a packet
of length L2. Then, the SU starts a vacation of length V2.
If the channel is busy, the SU starts a so-called virtual
transmission stage and then enters into the vacation stage
afterward. Here, virtual transmission means that the SU
does not actually transmit the packet but waits for a
time interval which is equal to the packet length. After
vacation, the SU senses the channel again.
•KS Scheme (Keep-Sensing-if-Busy): After a vacation,
the SU senses the channel. If the channel is idle, the
SU transmits a packet and then starts vacation. If the
SU senses the channel busy, it keeps sensing until the
channel is idle. Then, the SU transmits a packet and starts
a random vacation of length V2.
Since the sensing is perfect, given L2=τ, the collision
probability observed by the SU in the above random access
schemes is
Pr{Collision|L2=τ}=1−∞
τ
1
v1
e−t
v1dt
=1−e−τ
v1.
Note here we ignore the probability that one secondary packet
collide with multiple PU’s busy periods. This is reasonable
when l2is much smaller than l1+v1. However, we note that
in all simulations, such events are not ignored. We show that
the analysis and simulation match well for a wide range of
values.
In the VX scheme, the transmission activity (including
virtual transmission) of the SU is independent of the PU’s
occupancy of the channel, thus its analysis is simplified. In
this paper, we obtain closed form analysis on the collision
probability, the overlapping time, the capacity of the SU.
The closed-form solutions provide insights on the system
performance and facilitate the implementation of the MAC
protocol. On the other hand, the analysis is more difficult in
the KS scheme, since the transmission of the SU is somewhat
dependent on the activities of the PU. Interesting enough,
simulation results show that the throughput performance of the
KS scheme is indistinguishable from that of the VX scheme
under the same collision probability constraint.
B. Performance Analysis of VX Scheme Under Collision Prob-
ability Constraint
In the VX scheme, the probability that the SU actually
transmits a packet is equal to the probability that the SU senses
the channel idle. Due to the independence between the sensing
activities of the SU and the activities of the PU, this probability
is α=v1/(v1+l1). Let fL2(τ)be the probability density
function of L2. The average collision probability observed by
4
the SU can be expressed as below:
Pc
2=∞
0
(1 −e−τ
v1)fL2(τ)dτ. (5)
The virtual transmission stage in the VX scheme has the
same statistic characteristics as the actual transmission. Thus,
the number of “collision” events in time interval [0,T]can
be calculated as α·Pc
2·T
l2+v2. Consequently, the collision
probability for the PU can be calculated as
Pc
1= lim
T−>∞
α·Pc
2·T
l2+v2
T
l1+v1
=Pc
2
v1
l2+v2
.
(6)
Let ˆ
L2denote the contribution of each transmission to the
throughput of the SU. In particular, if a SU’s packet of length
L2is successfully transmitted, ˆ
L2=L2, otherwise ˆ
L2=0.
Let ˆ
l2=E[ˆ
L2]. We can regard ˆ
l2as the average length of
effective packets. Then we have
ˆ
l2=∞
0
1
v1
e−t
v1t
0
τfL2(τ)dτdt
=∞
0
τe−τ
v1fL2(τ)dτ.
(7)
The achievable capacity of the SU is obtained as the time
proportion that the SU actually transmits packets (excluding
the time when the SU performs virtual transmission) without
collision, i.e.,
C2=αˆ
l2
v2+l2
=α∞
0τe−τ
v1fL2(τ)dτ
v2+l2
.(8)
Our objective is to find the optimum l2,fL2, and v2to
maximize C2in (8) under the collision probability constraint
Pc
1≤η.
According to (6), to satisfy the collision probability con-
straint Pc
1≤η,wehave
l2+v2≥v1∞
0(1 −e−τ
v1)fL2(τ)dτ
η.(9)
Note that in (9), when l2v1+l1,wehavev2>0since
the right hand side in (9) can be approximated as l2/η. The
maximum throughput of the SU is achieved when equality
holds in the above inequality. Thus, we have
Cmax
2=ηα ∞
0
τ
v1e−τ
v1fL2(τ)dτ
∞
0(1 −e−τ
v1)fL2(τ)dτ .(10)
For the VX scheme, we have the following result.
Proposition 1. For a primary channel of which the idle
period V1obeys the exponential distribution, suppose that the
probability of channel being idle as α, under the constraint
that the collision probability of the primary user is less than
η, there exists an upper bound on the achievable capacity of
the SU which is expressed as below:
C2≤η·α. (11)
Proof: We have
∞
0
τ
v1
e−τ
v1f(τ)dτ +∞
0
e−τ
v1fL2(τ)dτ
=∞
0
(τ
v1
+1)e−τ
v1fL2(τ)dτ
≤∞
0
(τ
v1
+1) 1
1+ τ
v1
fL2(τ)dτ
=1,
where the inequality holds because e−x<1
x+1 for x>0.
Thus,
∞
0
τ
v1
e−τ
v1fL2(τ)dτ ≤1−∞
0
e−τ
v1fL2(τ)dτ .
Therefore, we have
∞
0
τ
v1e−τ
v1fL2(τ)dτ
∞
0(1 −e−τ
v1)fL2(τ)dτ ≤1
As a result,
C2≤ηα.
Note that, through the derivation, we only require that
the primary idle period be exponential and do not pose any
requirement on the distribution of L1,L2, and V2.
The result is somewhat surprising: C2cannot be larger
than ηα to satisfy the collision probability constraint, where
αis the time fraction of spectrum vacancy. The intuition is
as follows. Since the idle period is exponentially distributed,
whenever an SU starts to transmit on an idle channel, it faces
the same probability of collision. Therefore,the aggressiveness
of an SU’s transmission should be proportional to the collision
probability allowed by PUs. When η=1, the SU can fully
utilize the idle periods, and thus reach the capacity limit α
when l2goes to zero. On the other hand, if η=0, the SU can
never transmit and thus the capacity is zero. We note that in
both access schemes, the aggressiveness is controlled by L2
and V2.
The other side of the story is more straightforward. If the
idle period is deterministic, the capacity of SU, C2, can reach
αregardless of the value of η. An SU can simply track the
beginning of the idle period and transmit until the end of
the idle period. Deterministic and exponential distribution are
the two extremes in terms of predictability/entropy. While one
can maximize C2in the case of deterministic idle period, our
conjecture is that the exponential idle period results in the
worst opportunistic spectrum access capacity among all idle
period distributions (for a given l1and v1).
In the following, we consider two special cases, i.e., expo-
nentially distributed L2and fixed L2.
1) Exponentially distributed L2:When the packet length
of the SU, L2, is exponentially distributed, we have
Pc
2=l2
l2+v1
,(12)
5
Pc
1=v1
v2+l2
·l2
l2+v1
.(13)
Following (7), we have
ˆ
l2=l2
v2
1
(l2+v1)2.(14)
From (9), in order to satisfy the collision probability, for a
given l2, the optimal v2should be chosen such that
v2=max{0,v1l2
η(v1+l2)−l2}.(15)
Therefore, for given l2and η,Cmax
2is given as:
Cmax
2=ηα v1
l2+v1
.(16)
We can observe that the smaller l2, the larger the C2. This is
intuitive. With smaller l2, the collision probability is smaller,
and the amount of transmission wasted is smaller when a
collision happens. Therefore, more packets can be transmitted
successfully with the collision constraint satisfied. We note
that Cmax
2→ηα when l2→0.
2) Fixed Packet Length of SU: If the SU uses fixed packet
length, i.e., L2=l2,wehave
Pc
1=(1−e−l2
v1)v1
v2+l2
,(17)
From (9), in order to satisfy the collision probability, for a
given l2,v2should be chosen such that
v2=max{0,v1(1 −e−l2/v1)
η−l2}.(18)
Following similar approach in (8), we have
Cmax
2=ηα l2e−l2/v1
v1(1 −e−l2/v1).(19)
Again, we can observe that Cmax
2→ηα when l2→0.
3) Choosing fixed length packet: It is interesting to compare
the capacity performance of the SU with random and fixed
length packet (with the same mean). We have the following
Proposition for the VX scheme.
Proposition 2. For VX, let the largest packet length be lmax =
(2 −l2/v1
el2/v1−1)v1. Under the constraint Pc
1≤ηand E[L2]=
l2, the SU achieves the maximum throughput when it transmits
fixed length packets, i.e., L2=l2. In other words,
C2(l2)=ηα ∞
0
τ
v1e−τ
v1fL2(τ)dτ
∞
0(1 −e−τ
v1)fL2(τ)dτ ≤ηα
l2
v1e−l2/v1
1−e−l2/v1,
(20)
where fL2(τ)is the probability density function of L2with
finite support 0≤τ≤lmax.
Proof: Define X=L2
v1and its expectation mX=l2
v1.
From our assumption of fL2(τ), we know that the probability
density function of X,fX(x)=0if x∈ [0,2−l2/v1
el2/v1−1].
Next, we define an auxiliary function:
g(x)=(emX−1)xe−x+mXe−x,0<x<2−mX
emX−1,
whose second order derivative is
g(x)=(x(emX−1) −(2emX−2−mX))e−x.
For 0<x<2−mX
emX−1,g(x)≤0; thus, g(x)is concave
over the support of fX(x). Therefore, by Jensen’s inequality,
we have:
E[g(X)] = ∞
0
g(x)fX(x)dx
≤g(E[X]) = g(mX)=mX,
or,
∞
0
[xe−x(1 −emX)+e−mXmXe−x]fX(x)dx ≤e−mXmX.
Simple manipulations of this equation leads to the result of
the proposition:
∞
0xe−xfX(x)dx
∞
0(1 −e−x)fX(x)dx ≤mXe−mX
1−e−mX.
Proposition 2 specifies a maximum packet length. First,
packet size is limited in practice. Second, we note that, if L2>
v1, the collision probability will be very high and undesirable
for both the PU and SU. Since lmax =(2−l2/v1
el2/v1−1)v1>v
1,
lmax is a reasonable length constraint.
4) Simulations: In our simulations, we set v1=1,η=
0.1, and l1=0.5, leading to the channel idling probability
α=0.667 which approximately equals the proportionof white
space according to measurement.
In Fig. 2, we present results for the VX scheme when the
SU adopts exponentially distributed packets and fixed length
packets, respectively. For 106busy PU periods, we vary the
average SU packet length l2from 0.1to 1.0. Parameter v2of
the SU is obtained according to (15) and (18) to satisfy the
collision probability constraint. With no assumption about the
distribution of L1and V2in the analysis, different distributions
are tested to verify the analytical results. Here, we include the
cases L1being exponentially distributed (denoted by E) and
fixed (denoted by F). The distributions of V2are exponential
(denoted by E) and uniform over [0,2v2](denoted by U). In
the legend, X/Y means that L1follows X distribution and V2
follows Y distribution. Simulation result matches our analysis
very well, both in terms of collision probability and capacity
for different distributions of L1and V2.
In Fig. 3, we compare the throughput of VX and KS.
Without a selection criterion of v2for the KS, for each given
l2,wesetv2for the KS scheme using the same formula for the
VX scheme with η=0.1. We then adjust the value of v2in
the VX scheme using the actual collision probability obtained
from the simulation results of the KS scheme, and compare
the throughput. Hence, the comparison is fair because they
achieve the same collision probability. We observe a higher
collision probability for the more aggressive KS scheme. Also,
for both access schemes, the SU with fixed length packet
always achieves larger throughput.
6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.03
0.04
0.05
0.06
0.07
Throughput of SU: C2
C2 with VX, E/E
C2 with VX, E/U
C2 with VX, F/U
Analytical C2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1
0.105
0.11
Average Packet length of SU: l2
Collision Probability: P1
c
P1
c, E/E
P1
c, E/U
P1
c, F/U
η
l1 = 0.5
v1 =1
Fixed L2
Exponential L2
Fig. 2. Throughput of SU in VX scheme.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.04
0.05
0.06
0.07
0.08
0.09
Throughput of SU: C2
C2 with VX
C2 with KS
Analytical C2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.11
0.12
0.13
0.14
0.15
Average Packet Length of SU: l2
Collision Probability: P1
c
VX with fixed L2
KS with fixed L2
VX with exp. L2
KS with exp. L2
Fixed L2
Exponential L2
l1 = 0.5
v1 = 1
Fig. 3. Compare VX and KS schemes.
5) Observations:
•VX and KS schemes have indistinguishable throughput
for the SU under the same collision probability constraint.
Therefore, insistent sensing in KS scheme does not help.
The main reason is that the idle period of channel, V1,
is exponentially distributed and the SU has to guarantee
the collision probability.
•For VX scheme, an upper-bound of the throughputof the
SU is C≤ηα. We conjecture that this upper-bound is
valid for any access schemes that exploit the idle time of
a memoryless channel without coordination from the PU
under the collision probability constraint.
•For a large range of packet length l2, fixed length
packet achieves the best capacity overother packet length
distributions.
C. Overlapping Time Constraint
The upper bound of the SU throughputC2≤ηα may appear
pessimistic, but not surprising. Consider applications with
l2l1. For example, when the primary has many packets in
each busy session (e.g., a large file transfer) or the primary has
voice traffic on the channel, one secondary packet may overlap
with a very small proportion of PU transmission in a busy
period. Most PU packets will be successful while only a small
portion suffers from collision. Thus, the collision probability
constraint on Pc
1is overly pessimistic because it counts the
whole busy period as collision. To be more practical, we
introduce the overlapping time constraint as defined in (2),
and study the corresponding throughput performance of the
SU.
Our objective is to relate the overlapping time constraint to
the previously discussed collision probability constraint. The
problem is to calculate Pr
1in (2). Denote Lvas the random
length of overlapping time given that there is a collision. We
have:
Pr
1=Pc
1
E[Lv]
l1+v1
.(21)
Under the assumption that l2l1+v1, the probability that the
packet from the SU collides with more than one busy period
of the PU is negligible. Therefore, Lv=min{L1,L
residual
2},
where Lresidual
2is the remaining transmission time of the SU’s
packet when the PU returns to the channel. For brevity, we
present two special cases here.
First, when L2is exponentially distributed, Lresidual
2fol-
lows the same distribution as L2.IfL1is exponential with
mean l1, then Lvis also exponential with mean l1l2
l1+l2. Con-
sequently, we have
Pr
1=Pc
1
l1l2
(l1+l2)(l1+v1).(22)
From (22), we can observe that Pr
1increases linearly with
respect to Pc
1and the increasing factor is much smaller than
1. This is reasonable since the overlapping part is only a very
small proportion of the whole busy period if l2l1.
If L2=l2(fixed length packet) and L1is exponentially
distributed, we have
E[Lv]=l1−l2
1(e−l2/v1−e−l2/l1)
(v1−l1)(1 −e−l2/v1).(23)
Consequently, the proportion of the overlapping time is
Pr
1=Pc
1
l1[v1(1 −e−l2/v1)−l1(1 −e−l2/l1)]
(l1+v1)(v1−l1)(1 −e−l2/v1).(24)
Based on (21), (22), and (24), we can easily convert the
optimization problem under the overlapping time constraint
to a problem under the corresponding collision probability
constraint. Therefore, most of previous results in solving the
collision probability constraint apply to this case.
In our simulations, we set l1=0.5,v1=1, and
l2=0.05 = 0.1l1. The average vacation time v2is set to
satisfy the collision probability constraint Pc
1≤η. Fig. 4
shows the resulting Pr
1with respect to ηfor exponentially
distributed L2and fixed L2. We can observe that, in both cases,
Pr
1increases linearly with ηand the analytical results agree
with simulations. Also observe that for fixed L2, the fraction
of overlapping time is smaller than that with exponentially
7
distributed L2for any given η. This again shows the advantage
of fixed SU packet length. Our results are much more opti-
mistic by using overlapping time constraint. In particular, the
achievable throughputfor the SU is much closer to the limit of
the available spectrum holes, α. For example, for exponentially
distributed L2, if the constraint is posed as Pr
1≤0.018, the
throughput of the SU is C2=0.4. When the constraint is
Pr
1≤0.015, the throughput of using fixed length packet is
C2=0.65, very close to the maximum α=0.67.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.02
0.04
0.06
Collision Probability: P1
c
Overlapping Proportion: P1
r
Simulated P1
r for fixed L2
Analytical P1
r for fixed L2
Simulated P1
r for exp. L2
Analytical P1
r for exp. L2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.5
1
Throughput of SU: C2
C2 for fixed L2
C2 for exp L2
l1 = 0.5
v1 = 1
l2 = 0.05 Exponential L2
Fixed L2
P1
r = 0.01, fixed L2
P1
r = 0.018, exp. L2
P1
r = 0.016, fixed L2
P1
r = 0.0030, exp L2
Fig. 4. Throughput of SU under different constraints.
D. Packet Overhead
Our results have thus far favored smaller l2for better
throughput C2. This requires quick actions by SU. However,
packet overhead must be considered in practical transmission
by the SU. Therefore, we consider the effect of packet over-
head here by assuming that a fixed length overhead is added to
a payload of length L2. Let l0be the length of the overhead.
The collision probability observed by the SU is:
Pc
2=Pr{l0+L2>V
1}
=∞
0
(1 −e−l0+t
v1)fL2(t)dt. (25)
Correspondingly, the PU collision probability constraint is
Pc
1=Pc
2·v1
l2+v2+l0
≤η. (26)
The average length of successfully transmitted payload of the
SU is
ˆ
l2=∞
0
tfL2(t)∞
t+l0
1
v1
e−τ
v1dτdt
=e−l0
v1∞
0
te−t
v1fL2(t)dt.
(27)
To satisfy the collision probability constraint, we have the
optimal v2for the VX scheme as:
v2=max{0,v1Pc
2
η−l2−l0}(28)
Thus, the effective throughput of the SU is
C2=αˆ
l2
l2+v2+l0
≤αη ∞
0
t
v1e−l0+t
v1fL2(t)dt
∞
0(1 −e−l0+t
v1)fL2(t)dt
.
(29)
Using a similar approach as in the proof of Proposition 1,we
can show that Cmax
2<αη. In this case, small packet size may
not always be desirable. Next, we present the optimal packet
size when L2is exponential and fixed, respectively.
1) Exponentially distributed L2:We start with
Pc
1=α1−v1
l2+v1
e−l0
v1l1+v1
l2+v2
.(30)
The maximum effective throughput (excluding the overhead)
of the SU is:
Cmax
2=αη l2v1
(l2+v1)2
e−l0/v1
1−v1
l2+v1e−l0/v1.(31)
The optimal average packet length l2is:
l∗
2=v11−e−l0
v1.(32)
2) Fixed Length L2:For fixed payload length of the SU,
Pc
1=(1−e−l0+l2
v1)v1
v2+l2+l0
,(33)
Cmax
2=ηα l2
v1
e−(l0+l2)/v1
1−e−(l2+l0)/v1.(34)
The optimal l2is the solution of
1−l∗
2
v1
−e−l∗
2+l0
v1=0,(35)
In Fig. 5, we illustrate the numerical results of the through-
put performance of the VX scheme with exponentially dis-
tributed L2and fixed L2. In the calculation, we set l1=0.5,
v1=1,l0=0.05, and η=0.1. Clearly, the throughput of the
SU with overhead is smaller than that without overhead. The
optimal l2for both exponentially distributed L2and fixed L2
demonstrate the trade-off between overhead cost and packet
collision. The optimal values of l2in this simulation are found
to be l∗
2=0.21 and l∗
2=0.26 for exponential L2and fixed
L2, respectively.
V. RANDOM ACCESS SCHEME FOR MULTI-BAND
COMPETITIVE SYSTEM
Now consider the system with multiple primary channels
and multiple SUs. Denote the number of channels N, and
the number of SUs M. Let only one PU own each channel.
Each SU can only transmit in one channel at a time. Multiple
SUs compete for available spectrum in Nchannels. We
consider the VX scheme only in this section. All SUs adopt
the same access parameters, and thus they can be viewed
as homogeneous. They do not communicate with one other.
Additionally, as perfect sensing is assumed, an SU can detect
8
0 0.2 0.4 0.6 0.8 1
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Average Packet Length: l2
Throughput of SU: C2
exp. L2
fixed L2
exp. L2 overhead
fixed L2 overhead
l1 = 0.5
v1 = 1
overhead l0 = 0.05
Fig. 5. Throughput of SU with Packet Overhead.
transmissions by both the PU and other SU’s in a channel.
Assuming instantaneous sensing, as in Section IV, collision
in channels can only happen between the returning incumbent
PU and a transmitting SU.
Two sensing strategies are considered:
•Random-Sensing: After a random vacation time V2, each
SU randomly selects a channel, and then detects whether
the channel is busy. If it is, then SU enters the Virtual
Transmission stage. If the channel is idle, the SU trans-
mits its packet before taking a vacation.
•All-Channel-Sensing: After a vacation, eachSU senses all
channels. If there is no idle channel, the SU enters the
Virtual Transmission stage. Otherwise, the SU randomly
selects an idle channel for packet transmission.
With the Random-Sensing strategy, the SU only needs to
monitor one band at each instant. By comparison, the All-
Channel-Sensing strategy requires that each SU monitor all
channels. Thus, the former is much easier to implement than
the latter.
We present Monte-Carlo simulations on the performance
of the two strategies. We set l1=0.5,v1=1, and
l2=0.1. Due to limited paper length, we only present
results for exponentially distributed L2here. The aggregated
SU throughput is defined as the sum throughput of all SUs
in one particular channel. Similarly, the aggregated collision
probability is the collision probability observed by the PU. For
comparison, we introduce a One-Band-One-Secondarysystem
(OBOS), where the SU has the same average packet length l2
and the collision constraint of the PU is equal to the aggregated
collision probability in the multi-band competitive system.
Fig. 6 illustrates the aggregated throughput of MSUs and
the collision probability of the PU when N=1. We can see
that, for fixed l2and under the same collision probability, the
aggregated throughput of MSUs is the same as the throughput
of the SU in an OBOS system. In other words, given the
same collision probability constraint, the system with multiple
SUs has no throughput loss/gain. This is reasonable, because
there is no collision between SUs under perfect sensing.
Since SUs are homogeneous, each SU achieves an equal
fraction (1
M)of the total throughput. We can also observe
that collision probability caused by one individual SU with
M>1is less than the collision probability with M=1.
This is due to the lower probability of the channel being
idle from the perspective of one SU. Additionally, each SU
contributes proportionally to the collision probability of the
PU, demonstrated by the almost linear increase of Pc
1with
respect to M.
1 2 3 4 5 6
0.02
0.04
0.06
0.08
0.1
0.12
Aggregated Throughput
Simulated Aggregated C2
Analytical C2 for M=1, N=1
1 2 3 4 5 6
0
0.05
0.1
0.15
0.2
Number of SUs: M
Aggregated P1
c
Aggregated P1
c
l1 = 0.5
v1 = 1
l2 = 0.1
v2 = 2.2
N = 1 C2 = 0.0822
P1
c = 0.1351
Fig. 6. Aggregated Throughput of SUs in VX scheme.
Next, we test a more general case where there are MSUs
and Nprimary bands with M=3N. Note that the primaries’
activities are i.i.d., and all SUs behave in the same way, the
performance is the same for all channels. Therefore, we only
show the results for one of Nchannels here. The aggregated
throughput of SUs and collision probability in each chan-
nel for Random-Sensing and All-Channel-Sensing strategies
are shown in Fig. 7. The results show that, the aggregated
throughput of SUs for both sensing strategies matches very
well with the throughput in the OBOS system under the
condition that they have the same collision probability for
each channel. If we adjust the values of l2and v2, such
that the aggregated collision probabilities caused by Random-
Sensing and All-Channel-Sensing strategies are the same, then
they will have the same throughput. This indicates that, All-
Channel-Sensing strategy does not improve the total spectral
efficiency, despite the added complexity. This is mainly due
to the memoryless characteristics of the idle time, rather than
the limitation that each SU can access one channel each time.
We also observe that, without dividing the available bands
explicitly among multiple SUs, the autonomous random access
performs the same as the coordinated method of organizing
SUs into separate groups, each assigned a group of spectral
bands. From the perspective of each secondary user, the total
throughput it can achieve is N
MN=N
3times the aggregated
throughput in each channel.
9
1 2 3 4 5 6 7 8
0.1
0.15
0.2
Aggregated Throughput
Random−Sensing
OBOS with Random−Sensing’s P1
c
All−Sensing
OBOS with All−Sensing’s P1
c
1 2 3 4 5 6 7 8
0.1
0.15
0.2
0.25
Number of Bands:N
Aggregated P1
c
Random−Sensing
All−Sensing
l1 = 0.5
v1 = 1
l2 = 0.1
M = 3N
Fig. 7. Aggregated Throughput of SUs with Multiple Primary Bands.
As a summary for multi-band multi-user systems, we have:
•For the same collision probability constraint, the system
with multiple SUs has no loss/gain in terms of total
throughput.
•Under the same collision probability constraint, sens-
ing all the frequency bands does not improve the total
throughput of SUs.
•Dividing SUs into groups to access partitioned bands has
the same throughput as the strategy of allowing each SU
to randomly access all bands.
VI. CONCLUSIONS AND FUTURE WORK
In this paper, we study the data capacity of cognitive radio
users in opportunistic spectral access under stringent intru-
sion constraints on collision probability and the overlapping
collision time. Two random access schemes are proposed for
cognitive radio devices to exploit the spectral opportunities
in primary bands. We obtain closed-form expressions for the
collision probability of the PU and the capacity of the SU.
We show that the proposed random access schemes have
similar capacity performance. Furthermore, we find that the
collision probability and the overlapping time constraints are
closed related. SUs can utilize a significant amount of spectral
vacancy in the primary band under overlapping time constraint
when appropriate. In addition, we consider the overhead cost
in the SUs’ packets and demonstrate that an optimal trade-
off can be obtained for exponential and fixed packet length.
Finally, we investigate the aggregated throughput performance
and collision probability in a multi-band multi-secondary
system.
Our work provides a better understanding on the fun-
damental properties and performance limit of opportunistic
spectrum access. Future works in this direction may involve
the extension to systems with inaccurate sensing devices.
Imperfect sensing will induce collisions between different SUs
and hidden nodes problems in networks. Advanced signal
processing techniques in physical layers can also be integrated
in the design of media access schemes.
VII. ACKNOWLEDGMENT
This work is supported in part by the National Science
Foundation Grant CNS-0520126.
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