Analysis of cognitive radio networks with imperfect
sensing
Isameldin Suliman, Janne Lehtomäki and Timo Bräysy
Kenta Umebayashi
Centre for Wireless Communications (CWC)
University of Oulu
Oulu, Finland
Tokyo University of Agriculture and Technology (TUAT)
Tokyo, Japan
Abstract—Recently, cognitive radio access has received much
attention. Spectrum sensing methods are often used for finding
free channels to be used by cognitive radios (secondary users).
State diagram based approach can be used for analyzing the
effects of imperfect spectrum sensing (with false alarms and
misdetections). The state diagram consists of two-tuples like (1,2)
meaning one primary user and two secondary users present. We
note that state dependent transition rates are very important
for accurate modeling. This is because for example in state (3,0)
(all channels occupied by primary users) collisions happen with
increased probability. Our contribution is as follows. Explicit
expressions for state dependent transition rates are presented
for the case with three channels. However, the approach can be
used also for more channels. Primary termination probability
is used for evaluating the level of interference to primary users
caused by secondary users. Secondary success probability is used
to find out how often does a secondary call start and terminate
successfully. Simulation and analysis results agree very well.
I. I NTRODUCTION
Cognitive radio [1] has been considered as key mechanism
in addressing the problem of spectrum scarcity in wireless
communication systems [2]. Main system assumption in cognitive radio use is that there is a primary operator who owns (or
has licensed) the frequency band, and an opportunistic system
who attempts to use this band on a non-interfering basis,
i.e., as a secondary operator. Secondary users (i.e. users who
have associated with the opportunistic system) are users who
seeks free frequency channels for their own communication
purposes. Fig. 1 shows example of primary user occupancy in
N channels and also the free resource that could be utilized
for secondary user transmissions. Primary users (i.e. users
who are associated with the primary operator) however, have
strict priority over secondary users. This requires accurate
detection of the presence of the primary user by the secondary users. Different types of spectrum sensing mechanisms
have been used for detecting presence of primary users in
wireless networks. These mechanisms include energy [3] and
cyclostationary [4] based methods. For example, in energy
detection (see, e.g., [5]), the detector measures the energy /
power of the received signal during some time period and in
some frequency channel. Then the measured value of energy /
power is compared to a threshold. If the threshold is exceeded,
the detector decides that a primary signal was present. Perfect
detection of presence of primary users cannot be obtained by
978-1-4244-5213-4/09/ $26.00 ©2009 IEEE
Fig. 1.
Channel occupancy by primary users in N channels.
using spectrum sensing methods. There will be false alarms
and misdetections.
The interactions between primary and secondary users can
be studied by using continuous-time Markov chains (CTMC)
as in [6]. Therein, perfect sensing was assumed. The effects of
imperfect sensing can also be analyzed using Markov chains as
recently performed in [7]. Therein, a multichannel system was
assumed and the system state was reduced to two numbers: the
number of channels occupied by primary users and the number
of channels occupied by secondary users. State diagram based
approach was also used in [8] (with perfect radio resource
detection) for modelling dynamic spectrum access.
According to our best knowledge, in the existing literature
about Markov based analysis of cognitive radios, the state
dependence of events such as collisions with primary users
or finding free channels for secondary users is not fully taken
into account. In practice, the transition rates do not depends
only on the arrival/service rate but also on the system state.
For example, if there are several primary users present, then
the probability of collision is increased and also the probability
of finding a free channel is decreased.
In this paper, we apply Markov modeling approach similar
to [7], [8]. Our goal is to perform performance analysis of
a multichannel cognitive radio system with primary and secondary users and with imperfect sensing and random channel
search order. The performance metrics used are the probability
that a secondary user call is normally terminated and the
primary termination probability. The analysis is carried out
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by using the state diagram based approach. The prominent
feature of our state diagram is that it has state-dependent
transition rates at the nodes. These transition rates are found
by going through all the possible search sequences and by
taking into account the channel state. We believe that the
state dependency is very important for modeling cognitive
radio systems with higher accuracy. Explicit state diagram and
results are presented for case with three channels. However,
the approach can be used also for more channels.
PM : Misdetection => Collision
0.5
P
0.5
S
F
F
P
F
λ1
II. S YSTEM MODEL
P
False alarm probability PF refers to the probability that a
free channel is classified as being occupied. The misdetection
probability PM = 1 − PD is the probability that an occupied
channel is classified as vacant. The detection probability
PD is the probability that a occupied channel is correctly
classified as occupied. False alarms reduce spectrum utilization
of secondary users while misdetections cause interference to
the primary users.
We assume that there are N channels available. These
channels are shared between primary and secondary users,
with primary users having priority over the secondary users.
Calls of the primary users arrive with rate λ1 and secondary
calls arrive with rate λ2 . The corresponding service rates are
μ1 and μ2 . We have made some simplifications similar to those
used in [7] so as to enable finding theoretical solutions. The
assumptions are:
1) When primary user arrives to a channel occupied by
secondary users, the secondary user will always notice
the primary user [7]. Note that this results into a very
short collision with the primary user. After this, secondary user starts to search for a new channel. During
this phase, the secondary user will perform detection on
the remaining channels with random order until it finds
a free channel or all channels are determined to be busy.
Free channel is decided to be occupied with false alarm
probability PF and occupied channel is determined to
free with miss detection probability PM = 1 − PD ,
where PD is the detection probability
2) All state transitions are instantaneous, i.e., the time it
takes to search for a free channel is assumed to be
negligible. Note that in practice, the acquisition time can
be quite large, depending on the search method used [9].
3) A secondary user knows the channels occupied by other
secondary users and it will not use them. The necessary
information can be distributed over, for example, some
signaling channel.
4) A primary user knows the channels occupied by other
primary users so that there will be no collisions between
primary users [7]
5) In case of collisions (between primary user and secondary user), both colliding users withdraw from the
channel [7]. Note that the collision when primary comes
to secondary channel is assumed to be short and does
not cause the primary to leave the channel.
PF : False alarm
Fig. 3.
Movement from state (1,1) to state (1,0).
6) The search order for new free channels is random
(similar to random search in [9]). The search stops after
an idle channel is found or all channels are found to be
occupied. In practice, it might be allowed to search all
the channels several times until giving up.
III. A NALYSIS
We use a two-dimensional Markov chain to model the
system. The system states are given by two-tuples (i, j) where:
i is number of channels used for primary users’ calls and j
is number of channels used for secondary users’ calls. For
example (1,2) refers to state with one primary user and two
secondary users. Let N be the number of channels available
in the system. The total number of channel occupied by
primary and secondary users does not exceed N . Therefore,
we have the following restrictions: 0 ≤ i ≤ N , 0 ≤ j ≤ N ,
0 ≤ i + j ≤ N.
Let Q(i,j) denotes the steady state probability that the
system is in state (i, j), which can be interpreted as the
proportion of time that the system spends in state (i, j).
A. State transition diagram
Fig. 2 shows the state transition diagram when there are
three channels. The state-dependent transition rates have been
derived by going through all the possible sequences of channels and detection events. Some simple examples are given
next.
• Let us consider transition from state (1,1) to state (1,0).
The detection events and channel search orders that lead
to this transition are illustrated in Fig. 3. There are two
possibilities for this transition. In the first possibility, the
existing secondary user call ends (with rate μ2 ) so that
the number of secondary users is reduced by one. In
the second possibility, a new primary user comes to the
existing secondary user channel forcing it to search for
free channels. If the existing secondary user ends up with
colliding the existing primary user then both secondary
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3μ2
λ1
λ2 (1 − PF )
0, 3
λ1
3
(1 + 2 (1 − PF ))
0, 2
1, 2
P
λ 1 PM
4
F
P
D
1
λ1
λ1
1, 0
μ1 +
λ 2 PM
3
1 + PF + PF2
2μ1 +
Fig. 2.
2
μ2 + λ1 1 − PD
λ
2
λ2
0, 0
2
2μ1 + λ2 1 − PD
μ2 +
(1 + PF )
2 + PD + PF
+2PF PD
2
PF
1−PF
2, 1
3
3
(1 − PF ) (1 + PD )
2
μ2
1
2
PD
λ1
λ1
λ2 (1 − PF3 )
1+
1, 1
(1 + PF )
λ1
k
PD
λ 2 PM
2
λ1
2
k=0
μ1 +
2
2 + 1 − PF2
λ2 (1−PF )
3
λ2 (1−PF )
2
F
2μ2 + λ1 PM
2μ2
P
λ1
3
0, 1
(1 + PD )
2λ
3 1
PD
λ1
λ2 (1 − PF2 )
μ1 + λ2 PM
2, 0
λ 2 PM
3
(2 + PD + PF + 2PF PD )
3, 0
3
3μ1 + λ2 1 − PD
State transition probabilities
and primary calls will be terminated thus leaving only
the new primary thus resulting to the state (1,0). Let
us discuss this in more detail and start from the state
(1,1). Because primary knows about other primaries, the
new primary has two possible channels to use, one which
is occupied by secondary user and one which is totally
free. Therefore, the new primary user comes to secondary
user channel with probability 0.5 (primary is not concerned about secondary users). Then the secondary user
goes straight to the existing primary user channel with
probability 0.5 and collision happens with probability
PM . Alternative is that the secondary user goes first
to the free channel (with probability 0.5) and has false
alarm and then goes to the primary user channel and has
misdetection. All these terms result into rate of
μ2 +
•
λ 1 PM
(1 + PF )
4
Movement from state (2,1) to state (3,0). The new primary user always forces the secondary user to leave the
channel and to search new free channels. The secondary
user correctly detects that the two other channels are
2
. Thus the
occupied by primary user with probability PD
2
transition rate is λ1 PD .
B. Balance equations
The balance equations can be written by considering the
transition rates using the rule that input must equal output for
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each state [10]. Additionally,
N
N Q(i,j) = 1
Theory
Simulation
(1)
0.25
For example, for the simple case of the state (0,0) the
balance equation results into
0.2
Q(1,0) μ1 +
1 + PF + PF2 + Q(0,1) μ2
Q(0,0) =
(λ1 + λ2 (1 − PF3 ))
(2)
For the case of state (1,2) we have the following balance
equation
λ2 PM
3
Probability
i=0 j=0
0.1
0.05
Q(1,2) (λ
1 PD + 2μ2 + λ1 PM + μ1 + λ2 PM ) = Q(0,3)
λ1 +
Q(1,1)
λ2 (1−PF )(1+PD )
3
+ Q(0,2)
0.15
0
λ1 (1+2(1−PF ))
3
(3)
The resulting set of simultaneous linear equations (balance
equations for every state and the normalization constraint) can
be easily solved using, for example, MATLAB.
1
2
3
4
5
6
State index
7
8
9
10
Fig. 4. State probabilities, simulation versus theory, PF = 0.15, PD =
0.713, λ1 = 7, λ2 = 3.5, μ1 = μ2 = 4, N = 3 channels.
C. Primary user termination probability
PP T
⎡ N N
−i
(i,j)
Q(i,j) T(i−1,j) − iμ1
⎢ i=1 j=0
⎢
−1 N
−i
⎣ N
(i,j)
+
Q(i,j) T(i,j−1) − jμ2
i=1 j=1
=
λ1 1 − Q(N,0)
⎤
⎥
⎥
⎦
(4)
0.25
Theory
Simulation
Primary termination probability
We use the term primary termination probability to refer to
the probability that a primary user call, which has not been
blocked in start, is terminated due to collisions with secondary
users because of misdetections. The probability that a primary
user call is terminated due to collisions with secondary users
can be found by going through the state diagram states. The
result is
0.2
0.15
0.1
0.05
(i,j)
where Q(i,j) is the state probability of state (i,j) and T(i−1,j)
is the transition from state (i,j) to (i − 1,j).
0
0.6
IV. N UMERICAL AND S IMULATION RESULTS
First, we performed simulation to verify the solution to the
state equations. The simulations were performed with MATLAB using an event-based approach and Poisson arrival processes. The simulation setup used the assumptions mentioned
in Section II, i.e., the acquisition time was negligible. However, during acquisition the channels were randomly searched
using the specified false alarm and detection probabilities,
i.e., we did not simulate the Fig. 2 directly thus providing
verification for the derived state transition probabilities. The
theoretical results are compared with simulation in Fig. 4 for
the case N = 3. It can be seen that the theory and simulation
agree very well. The state numbers are explained in Table I.
Fig. 5 shows the primary termination probability (theory and
simulation). Of course, when PD = 1, the primary termination
probability is zero. From the results we can see that if 5 %
0.65
0.7
0.75
0.8
PD
0.85
0.9
0.95
1
Fig. 5. Primary termination probabilities, simulation versus theory, PF =
0.15, λ1 = 7, λ2 = 3.5, μ1 = μ2 = 4, N = 3 channels.
is the maximum allowed termination probability (caused by
secondary users interference), then the detection probability
PD must be around 0.95 (or greater).
Fig. 6 shows the probability that a secondary call is started
and terminated normally. It can be seen that as expected the
success probability goes down when λ1 increases. This means
that the high primary arrival rate means that the channels are
more often occupied by primary users reducing opportunities
for secondary users to access the network.
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R EFERENCES
Probability that a secondary call is succesfull
1
Theory
Simulation
0.9
0.8
0.7
μ =4
2
0.6
μ2=8
0.5
0.4
μ2=2
0.3
0.2
0.1
0
1
2
3
4
5
λ1
6
7
8
9
10
Fig. 6. Probability that a secondary user call is normally terminated, PF =
0.15, λ2 = 3.5, μ1 = 4, N = 3 channels.
[1] J. Mitola III, “Cognitive radio: An integrated agent architecture for software defined radio,” Ph.D. dissertation, Royal Institute of Technology,
Sweden, 2000.
[2] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Select. Areas Commun., vol. 23, pp. 201–220, Feb. 2005.
[3] H. Urkowitz, “Energy detection of unknown deterministic signals,” Proc.
IEEE, vol. 55, no. 4, pp. 523–531, Apr. 1967.
[4] W. A. Gardner, “Signal interception: a unifying theoretical framework
for feature detection,” IEEE Trans. Commun., vol. 36, no. 8, pp. 897–
906, Aug. 1988.
[5] J. Lehtomäki, “Analysis of energy based signal detection,” Ph.D.
dissertation, Acta Univ Oul Technica C 229. Faculty of Technology,
University of Oulu, Finland, Dec. 2005. [Online]. Available:
http://herkules.oulu.fi/isbn9514279255/
[6] B. Wang, Z. Ji, M. Abdulrehem, R. Liu, and T. Clancy, “Primaryprioritized markov approach for dynamic spectrum allocation,” IEEE
Trans. Wireless Commun., vol. 8, pp. 1854–1865, Apr. 2009.
[7] S. Tang and B. L. Mark, “Modeling and analysis of opportunistic spectrum sharing with unreliable spectrum sensing,” IEEE Trans. Wireless
Commun., vol. 8, pp. 1934–1943, 2009.
[8] Y. Xing, R. Chandramouli, S. Mangold, and S. S. N., “Dynamic
spectrum access in open spectrum wireless networks,” IEEE J. Select.
Areas Commun., vol. 24, no. 3, pp. 626–637, 2006.
[9] L. Luo and S. Roy, “Analysis of search schemes in cognitive radio,” in
SECON, 2007, pp. 647–654.
[10] L. Kleinrock, Queueing Systems, Volume I: Theory. John Wiley and
Sons, Mar. 1975.
TABLE I
M APPING BETWEEN (i,j) AND STATE INDEX
(i,j)
(0,0)
(1,0)
(2,0)
(3,0)
(0,1)
(1,1)
(2,1)
(0,2)
(1,2)
(0,3)
state index
1
2
3
4
5
6
7
8
9
10
V. C ONCLUSIONS AND FUTURE WORK
We have presented analysis of cognitive radio networks
with imperfect sensing. Our approach employed a multidimensional Markov chain (state diagram) to obtain exact theoretical
probabilities. We used state dependent transition rates provide
more accurate analysis. As performance metrics we used the
probability that a primary call is terminated by secondary
users due to misdetections and the probability that a secondary
call is successful. The main results of the paper are that the
probability of collision to primary users increases with the
probability of miss-detection and that the probability of successful secondary communications decreases with the primary
traffic arrival rate. In the future work, it would be useful to
investigate the case where secondaries will not always notice
the arrival of the primary users and also the case where only
the secondary will leave when collisions occur. Additionally,
the state equations should be generalized to larger number of
channels and the effects of timing offset between primary user
arrival and secondary user sensing period could be studied.
The simulation setup could be extended to take into account
non-negligible acquisition times.
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