Performance Analysis of Cognitive Radio Spectrum Access With Prioritized Traffic

Abstract

Dynamic spectrum access (DSA) is an important design aspect for cognitive radio networks. Most of existing DSA schemes are to govern unlicensed user (i.e., secondary user, SU) traffic in a licensed spectrum without compromising the transmissions of the licensed users, in which all the unlicensed users are typically treated equally. In this paper, prioritized unlicensed user traffic is considered. Specifically, the unlicensed user traffic is divided into two priority classes (i.e., high and low priority). We consider a general setting in which the licensed users' transmissions can happen at any time instant. Therefore, the DSA scheme should perform spectrum handoff to protect the licensed user's transmission. Different DSA schemes (i.e., centralized and distributed) are considered to manage the prioritized unlicensed user traffic. These DSA schemes use different handoff mechanisms for the two classes of unlicensed users. We also study the impact of subchannel reservation for high-priority SUs in both DSA schemes. Each of the proposed DSA schemes is analyzed using a continuous-time Markov chain. For performance measures, we derive blocking probability, the probability of forced termination, call completion rate, and mean handoff delay for both high- and low-priority unlicensed users. The numerical results are verified using simulations.

Full text

Performance Analysis of Cognitive Radio Spectrum Access with
Prioritized Traffic
Vamsi Krishna Tumuluru, Ping Wang, and Dusit Niyato
Center for Multimedia and Network Technology (CeMNeT)
School of Computer Engineering, Nanyang Technological University, Singapore.
Abstract—Dynamic spectrum access (DSA) is an important de-
sign aspect for the cognitive radio networks. Most of the existing
DSA schemes are to govern the unlicensed user (i.e., secondary
user) traffic in a licensed spectrum without compromising the
transmissions of the licensed users, in which all the unlicensed
users are typically treated equally. In this paper, prioritized
unlicensed user traffic is considered. Specifically, we prioritize
the unlicensed user traffic into two priority classes (i.e., high
and low priority). Two different DSA policies are proposed to
manage the handoff for prioritized unlicensed user traffic (i.e.,
reassigning new channels to the secondary users which paved
way for licensed user). These two policies are different in which
one allows to drop the ongoing low priority secondary users to
accommodate more high priority secondary users being replaced
by the presence of licensed user while the other does not allow
that. We also study the impact of sub-channel reservation for
the high priority secondary users in both DSA policies. Both
DSA policies are analyzed using Markov chain. For performance
measures, we derive the blocking probability, the probability of
forced termination and the throughput for both high and low
priority unlicensed users. The numerical results are verified using
simulations.
Keywords –Cognitive radio, dynamic spectrum access,
Markov chain analysis, priority.
I. INT ROD UC TI ON
In recent years, several spectrum surveys have been con-
ducted to understand the spectrum utilization in the licensed
and unlicensed portions of the radio spectrum [1], [2]. These
surveys revealed that under the present spectrum regulatory
policies, major portions of the licensed spectrum are under-
utilized for most of the time, while the unlicensed spectrum
is heavily used and is often insufficient. Among the efforts to
improve the overall spectrum utilization, the concept of cogni-
tive radio is gaining much importance [3]. The cognitive radio
network (CRN) is composed of licensed users and unlicensed
users sharing the licensed spectrum. In most cases, the licensed
users (also called primary users) are oblivious of the existence
of the unlicensed users (also called secondary users). In the
CRN, the secondary users are allowed to dynamically access
unused channels (i.e., frequency bands) in the primary user
spectrum, thereby improving the overall spectrum utilization.
Secondary user transmissions in the CRN can be effectively
managed by a dynamic spectrum access (DSA) policy. Using
the DSA policy, the secondary users are assigned unused
channels in the primary user spectrum. In the event of a
primary user’s arrival, the secondary users’ transmissions on
the corresponding channel are either reassigned1to another
unused licensed channel or terminated. Recently, some works
dealt with the performance evaluation of the CRN under
different DSA policies, using continuous time Markov chain
(CTMC) analysis [4]-[7]. According to the DSA policy in [4],
the secondary users which experience unsuccessful handoff are
queued till they find the transmission opportunities. In [5],
DSA policy considered channel reservation for secondary user
handoff. In [6], the secondary user arrivals wait in a queue
when a channel2is not available. In [7], the DSA policy
assigns variable bandwidth to the secondary users. In all these
works, the CRN was modeled as a call admission control
system which receives calls from both primary users (PUs)
and secondary users (SUs). During the spectrum access, the
PU calls have a higher priority over the SU calls.
Unlike the above works, in this paper, we consider prioriti-
zation among the SUs while accessing the licensed spectrum.
For example, the SUs with real-time traffic have higher priority
than those with non real-time traffic. To date, only a few
works ([8] and [9]) have taken prioritized SU traffic into
consideration. However, these works have not addressed the
issue of spectrum handoff under prioritized SU traffic. For
the simplicity of presentation, we consider that the SU traffic
is composed of two priority classes3(i.e., high and low
priority). We propose two DSA policies to handle the spectrum
access for the SUs in the licensed spectrum. The two DSA
policies have different SU handoff mechanisms. Further, we
also introduce sub-channel reservation for the high priority SU
call arrivals in both the DSA policies. We develop analytical
models for the CRN under both the DSA policies using CTMC
models. The performance of the DSA policies is evaluated in
terms of the blocking probability, the probability of forced
termination, and the throughput for both high and low priority
SUs.
II. SY ST EM MO DE L
The licensed spectrum is divided into Mchannels, each
of which is further divided into Nsub-channels, as shown in
Fig. 1. A PU call is assigned one channel whereas a SU call is
assigned one sub-channel for data transmission. The licensed
1The process of reassigning a displaced secondary user transmission is
referred to as handoff.
2In this paper, the terms ‘licensed channel’ and ‘channel’ are used inter-
changeably.
3Please note that the proposed analytical model is not restricted to the two
priority class case and can be extended to the case of more than two priority
classes.
1

spectrum is shared by the PUs and SUs. The PUs have the
highest priority in accessing the channels. The secondary users
are classified into two priority classes. The high priority SUs
are denoted as SU1while the low priority SUs are denoted as
SU2. Similar to [4], we assume that a central controller exists
to implement the DSA policy. The objective of the DSA policy
is to assign idle sub-channels to the incoming SU calls, and
moderate their handoff. We consider two DSA policies for our
system model. They are denoted as DSA-C1 and DSA-C2.
Licensed spectrum
channel
MN
M
1 N
1
N(M-1)+1
sub-channel
Fig. 1. System model.
A. Sub-channel Reservation under DSA-C1 and DSA-C2
Under both the DSA policies, a number of sub-channels are
reserved for high priority secondary user SU1, i.e., when the
total number of idle sub-channels is less than ζwhich is the
number of sub-channels reserved for SU1, the new SU2call
will be rejected. In this way, we provide higher priority for the
SU1calls over the SU2calls4.
Call arrivals occur independently as Poisson processes with
mean arrival rates λp,λ1and λ2for PU, SU1and SU2, re-
spectively. The service times independently follow exponential
distributions with mean service rates of µp,µ1and µ2for PU,
SU1and SU2calls, respectively. The number of ongoing PU
calls is denoted as kand the number of occupied sub-channels
is denoted as Y. The new calls are dropped when the system
becomes full (i.e., Y=MN for SU1call, Y=M N −ζfor
SU2calls and kN =Mfor PU calls).
B. Handoff Mechanism under DSA-C1
When a new PU call claims a channel occupied by the SUs,
the handoff mechanism is initiated. During handoff, the idle
sub-channels (if any) are first assigned to the displaced SU1
calls. Thereafter, the remaining idle sub-channels are assigned
to the displaced SU2calls. If the required idle sub-channels are
not available for SU1handoff, then some ongoing SU2calls
are terminated and the resulting idle sub-channels are assigned
to the displaced SU1calls. When the idle sub-channels are not
enough to accommodate all the displaced SU1or SU2calls,
some of the displaced calls will be terminated.
C. Handoff Mechanism under DSA-C2
DSA-C2 policy is similar to DSA-C1 policy except that no
ongoing SU2calls are terminated for the sake of SU1handoff.
In other words, if a displaced SU1call does not find an idle
sub-channel, it is terminated.
4Note that the PUs are not affected by such sub-channel reservation.
III. PERFORMANCE ANALYSIS
In this section, we develop analytical models for the CRN
corresponding to each DSA policy using continuous time
Markov Chain (CTMC).
The state of the CTMC (for both DSA policies) is defined as
z= [i, j, k]. Here, i∈ {0,1, . . . , M N},j∈ {0,1, . . . , M N −
ζ}and k∈ {0,1, . . . , M }represent the number of ongoing
SU1,SU2, and PU calls in the system, respectively. The total
number of occupied sub-channels in the state zis calculated
as Y=i+j+kN . For a valid state, Yshould not
exceed MN. Let l(where l∈ {0,1, . . . , N }) and m(where
m∈ {0,1, . . . , N }) denote respectively the number of SU1
and SU2calls displaced by an incoming PU call when the state
is [i, j, k].land mshould satisfy the following conditions:
l≤i, m ≤jand l+m∈ {0,1, . . . , N }
r=MN −Y−(N−(l+m)) and r≥0
s= (l+m)−min(r, l +m).(1)
In Eq. (1), the first condition gives the maximum number of
SUs that can be displaced upon a PU arrival whereas the sec-
ond condition gives the number of sub-channels available for
handoff (denoted as r) for the displaced SU calls. Accordingly,
the total number of unsuccessful handoff calls (denoted as s)
is calculated.
The evolution of the state [i, j, k]of the CTMC is presented
under three cases of Y:
1) Y≤N(M−1): The system has at least Nidle sub-
channels.
2) N(M−1) < Y < MN : The system has idle sub-
channel in the range {1, . . . , N −1}.
3) Y=MN : The system has no idle sub-channel.
For the case Y≤N(M−1), all displaced SU1and SU2
calls perform successful handoff (i.e., s= 0) when displaced
by a PU arrival. In other words, no calls are terminated (under
both DSA-C1 and DSA-C2) due to an incoming PU call. For
the case N(M−1) ≤Y≤MN ,s={1, . . . , N −1}
number of SU calls experience unsuccessful handoff, whereas
when Y=MN ,s=NSU calls experience unsuccessful
handoff. The exact number of terminated SU1and SU2calls
under each centralized DSA policy are explained later. Let l0
and m0denote the number of terminated SU1and SU2calls,
respectively. Thus, l0+m0=s.
A. State Transitions under DSA-C1 Policy
Under the DSA-C1 policy, the transitions for the state
[i, j, k]for different cases of Yare explained in Fig. 2.
State transitions from/to the state [i, j, k]occur due to any
of the six possible events, namely PU arrival, SU1arrival,
SU2arrival, PU departure, SU1departure and S U2departure.
Each state transition is represented by its corresponding rate.
Taking as an example, a SU2arrival in state [i, j −1, k]
causes transition to state [i, j, k]with a rate δ1·λ2, where
δ1=1(i+(j−1)+kN <MN −ζ)specifies the condition for sub-
channel reservation (i.e., a new SU2call is accepted by the
system only when the condition i+ (j−1) + kN < MN −ζ
2

holds), where 1(·)is the indicator function which returns the
value 1when the condition given inside the parenthesis is true
and returns 0otherwise. In Fig. 2, δ2=1(i+j+kN <MN −ζ)
(condition for sub-channel reservation in state [i, j, k]) and
δ3=1(i+j+(k−1)N≤N(M−1)).
The transition rate γi,j,k
l0,m0from state [i, j, k]to state [i−
l0, j −m0, k + 1] is calculated as follows:
1) Let Nl0,m0denote the set containing the valid combina-
tions of (l0, m0).
2) For every valid pair of (l0, m0)in the set Nl0,m0, find
Rl0,m0which represents the number of valid combina-
tions of l,m,rand s=l0+m0. Valid l,m,rand sfor
given [i, j, k]found using Eq. (1).
3) Then, the transition rate γi,j,k
l0,m0is given by
γi,j,k
l0,m0=Rl0,m0
X
(l0,m0)∈Nl0,m0
Rl0,m0
λp.(2)
Taking into consideration the handoff mechanism explained in
Section II-B, the variables l0and m0in Eq. (2) are given by
l0=s−min(j, s)and m0= min(j, s). The value of sis
determined by Eq. (1).
p
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Fig. 2. State transitions for the DSA policies, DSA-C1 and DSA-C2.
For the cases Y≤N(M−1) and N(M−1) < Y < M N ,
all the transitions shown in Fig. 2 are valid. For the case Y=
MN , the transitions between the states [i, j, k]and [i+ 1, j, k],
and between [i, j, k]and [i, j + 1, k]are not considered as
the system is full (i.e., idle sub-channels not available). Apart
from the state transitions shown in Fig. 2, when Y=MN
few transitions occur from the states in which the number of
ongoing calls is greater than N(M−1) to the state [i, j, k]
due to a PU arrival. These transitions are described below.
•Transition from state [i, j+j0, k −1] to state [i, j, k]occurs
with rate γi,j+j0,k−1
0,j0, where j > 0and 0< j0< N . The
rate γi,j+j0,k−1
0,j0is calculated using Eq. (2) corresponding
to the state [i, j +j0, k −1]. During this state transition,
j0SU2calls are terminated.
•Transition from state [i+ (s0−j0), j +j0, k −1] to state
[i, j, k]occurs with rate γi+(s0−j0),j+j0,k−1
s0−j0,j0, where j= 0,
0< s0< N and 0≤j0< s0. The rate γi+(s0−j0),j +j0,k−1
s0−j0,j0
is calculated using Eq. (2) corresponding to the state [i+
(s0−j0), j +j0, k −1]. During this state transition, s0−j0
SU1calls and j0S U2calls are terminated.
B. State Transitions under DSA-C2 Policy
Fig. 2 also represents the state transitions under the DSA-C2
policy for the various cases of Y. According to the handoff
mechanism under DSA-C2 policy, the number of terminated
SU1and SU2calls are given as l0=l−min(r, l)and m0=
max(0, m −max(0, r −l)), respectively. All state transitions
shown in Fig. 2 occur similar to the DSA-C1 policy. Apart
from the state transitions given in Fig. 2, transitions occur from
states [i+i0, j +j0, k −1] (in which i+j+(k−1)N=N(M−
1)) to the state [i, j, k]when Y=M N , where 0≤i0, j 0< N
and 0< i0+j0< N.
C. Performance Measures
The performance measures for each DSA policy are ex-
pressed using the steady state probability distribution of its
corresponding CTMC. We derive performance measures (for
both SU1and SU2calls) such as blocking probability, proba-
bility of forced termination, and throughput.
Let Πzdenote the steady state distribution of a CTMC
whose state is denoted as z= [i, j, k]. To simplify the
presentation, we use the same notations under all DSA policies.
For any DSA policy, the corresponding steady state probability
distribution Πzis obtained by finding the corresponding tran-
sition rate matrix Qand applying the Gauss-Seidel algorithm
[10]. Each row of Qrepresents the transitions with respect to
a specific state z, as a balance equation. For example, referring
to Fig. 2, one balance equation with respect to the state [i, j, k]
under the DSA-C1 policy is expressed as follows:
[kµp+δ2λ2+λ1+λp+iµ1+jµ2]·Π[i,j,k]=
λpΠ[i,j,k−1] + (j+ 1)µ2Π[i,j+1,k]+ (i+ 1)µ1Π[i+1,j,k ]+
(k+ 1)µpΠ[i,j,k+1] +λ1Π[i−1,j,k]+δ1λ2Π[i,j−1,k ](3)
where Π[i,j,k]represents the steady state probability for state
[i, j, k]under DSA-C1 policy.
1) Blocking Probability: The blocking probability repre-
sents the probability that an incoming SU call (SU1or
SU2) is not permitted to enter into the system. The blocking
probability of SU1calls denoted as PB1is expressed as
PB1=P∀z
Y=MN Πz. The blocking probability of SU2calls
denoted as PB2is expressed as PB2=P∀z
Y≥MN −ζΠz.
2) Probability of Forced Termination: The probability of
forced termination represents the probability that an ongoing
SU call (SU1or S U2) is terminated by an incoming PU
call. The probability of forced termination for the SU1calls,
denoted as PF1, is expressed as follows:
PF1=X
∀z
i≥l0, j≥m0
l0·γi,j,k
l0,m0·Πz
λ1(1 −PB1).(4)
In Eq. (4), the numerator denotes the rate that l0SU1calls
are terminated in state zwhereas the denominator denotes the
3

effective rate with which a new SU1call is assigned a sub-
channel.
Similarly, the probability of forced termination for the SU2
calls, denoted as PF2, is expressed as follows:
PF2=X
∀z
i≥l0, j≥m0
m0·γi,j,k
l0,m0·Πz
λ2(1 −PB2).(5)
In Eqs. (4) and (5), l0and m0depend on the DSA policy.
3) Throughput: The throughput under a given SU priority
class is expressed as the mean number of ongoing calls in
the system. Let η1and η2denote the throughput for SU1and
SU2calls, respectively. The throughput η1is given by η1=
λ1(1 −PB1)(1 −PF1)whereas The throughput η2is given by
η2=λ2(1 −PB2)(1 −PF2).
IV. RES ULT S AN D DISCUSSION
In this section, we compare the two DSA policies based on
the performance measures described in Section III-C. The ac-
curacy of the analytical models is verified through simulations.
In the experiments, we set M= 3 and N= 5. The symbols
(a)and (s)in the figures indicate analytical and simulation
results, respectively.
A. Blocking Probabilities
The blocking probabilities corresponding to the DSA-C2
policy are same as that corresponding to the DSA-C1 policy
(because the DSA policies only differ in the handoff mech-
anism), and hence they are not shown for brevity of paper.
Fig. 3 shows the blocking probabilities of the SU1and SU2
calls under the DSA-C1 policy with various PU arrival rate
(λp). The following parameters are chosen for this experiment:
λ1= 0.4,λ2= 0.4,λp∈[0.03,0.12],µ1= 0.8,µ2= 0.8,
µp= 0.09, and η= 2. It can be seen that all the blocking
probabilities (i.e., for both SU classes) increase as the PU
arrival rate increases. This is because the number of busy
sub-channels increases with an increase in the PU arrival
rate, resulting in higher blocking probabilities for the SUs.
Fig. 3 also shows that the analysis results match well with the
simulation results.
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
PU arrival rate, λp
Blocking pro babilities of S U1and S U2calls
PB1, DSA-C1 (a)
PB1, DSA-C1 (s)
PB2, DSA-C1 (a)
PB2, DSA-C1 (s)
Fig. 3. Blocking probabilities of SU1and S U2calls under both DSA
policies.
B. Forced Termination Probabilities
Fig. 4 and Fig. 5 respectively show the forced termination
probabilities of the SU1and S U2calls under both DSA
policies with various PU arrival rate λp. The following pa-
rameters are set for this experiment: λ1= 0.8,λ2= 0.8,
λp∈[0.03,0.12],µ1= 0.35,µ2= 0.35,µp= 0.09
and η= 2. It can be observed that as λpincreases, the
forced termination probabilities of SU1and S U2calls also
increase with both DSA policies. Fig. 4 shows that DSA-
C1 policy has lower force termination probability for the
SU1calls compared to DSA-C2 policy. Fig. 5 shows higher
force termination probability of the SU2calls using DSA-
C1 policy compared to DSA-C2 policy. Such observations
accord with our expectation. Compared to the DSA-C2 policy,
DSA-C1 policy reduces handoff failures for the SU1calls by
terminating some ongoing SU2calls during handoff. Again,
the analysis and simulation results match well.
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
PU arrival rate, λp
Forced termina tion probab ility of SU1calls, PF1
DSA-C1 (a)
DSA-C1 (s)
DSA-C2 (a)
DSA-C2 (s)
Fig. 4. Forced termination probability of SU1calls under both DSA policies.
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
PU arrival rate, λp
Forced termina tion probab ility of SU2calls, PF2
DSA-C1 (a)
DSA-C1 (s)
DSA-C2 (a)
DSA-C2 (s)
Fig. 5. Forced termination probability of SU2calls under both DSA policies.
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
SU1arrival rate, λ1
Forced termina tion probab ility of SU2calls, PF2
DSA-C1 (a)
DSA-C1 (s)
DSA-C2 (a)
DSA-C2 (s)
Fig. 6. Effect of SU1arrival rate on forced termination probability of SU2
calls under both DSA policies.
4

Fig. 6 shows the effect of SU1arrival rate (λ1) on the forced
termination probability of the low priority SU2calls. In this
experiment, we set λp= 0.06 and vary λ1. It can be seen that
as λ1increases, the forced termination probability of SU2calls
also increases with both DSA policies. This is because when
λ1increases, the number of SU2calls entering the system
decreases. This leads to an increase in PF2along with λ1.
With DSA-C1 policy, apart from terminations caused by the
PU arrivals, the SU2calls are also terminated by SU1calls
during handoff. Thus, PF2of DSA-C1 policy is higher than
that of DSA-C2 policy.
C. Optimal Sub-channel Reservation
As mentioned in Section II-A, some sub-channels (ζ) are
reserved for the SU1call arrivals. Using the sub-channel
reservation, we block some low priority SU calls to improve
the performance of the high priority SU calls. Based on the
given parameter settings and blocking probability requirement
of the SU1calls, an optimal number of sub-channels (ζ)
to be reserved under the DSA policies can be determined
from our analysis. Here, optimal number means the minimum
number of sub-channels to be reserved in order to guarantee
that the blocking probability requirement of the SU1calls
is satisfied. For instance, consider the following parameter
setting: λ1= 1.8,λ2= 1.8,λp= 0.06,µ1= 0.8,µ2= 0.3,
µp= 0.09 and a desired blocking probability of 5% for the
SU1calls. From the analysis, we found that the optimal value
of ζis 4for both DSA policies. The analysis results can be
verified from simulation. Fig. 7 shows the simulation results
for the above parameter settings under different values of ζ.
Fig. 7 shows that the desired blocking probability of 5% for
the SU1calls is satisfied when ζ= 4 sub-channels.
0123456
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
Number of su b-cha nnels reserved for SU1calls (ζ)
Blocking prob ability of SU 1calls, PB1
DSA-C1 (s)
DSA-C2 (s)
Fig. 7. Effect of ζon blocking probability of S U1calls under both DSA
policies.
D. Throughput Evaluation
We verify the throughput calculated in the analysis with
the simulation results. For the simulation, the throughput for
a given SU priority class is calculated by taking the ratio of
the total number of SU calls corresponding to that priority
class completing service, to the total duration of the simulation.
In our simulations, call arrivals are generated for a duration
of 800000 time units. For the parameter setting λ1= 1.8,
λ2= 1.8,λp= 0.3,µ1= 0.3,µ2= 0.06,µp= 0.4and
ζ= 0, we obtained the following values for the throughput
from the analysis: η1= 0.9975 and η2= 0.3462 for DSA-C1
policy whereas η1= 0.7819 and η2= 0.3997 for DSA-C2
policy.
In the simulation, under DSA-C1 policy, 797929 SU1calls
and 277086 SU2calls completed service whereas under DSA-
C2 policy, 638938 SU1calls and 312457 SU2calls com-
pleted service. Therefore, from simulations, η1= 0.9974 and
η2= 0.3464 for DSA-C1 policy whereas η1= 0.7986 and
η2= 0.3906 for DSA-C2 policy. The analysis results and
simulation results for the throughput of the SUs correspond
closely. As expected, DSA-C1 policy gives higher throughput
for SU1compared to DSA-C2 policy, at the expense of
sacrificing the throughput of SU2. This phenomenon holds
under various parameter settings.
V. CONCLUSION
We have investigated the dynamic spectrum access in the
cognitive radio networks under a special case in which the SU
traffic is prioritized. Two different DSA policies have been
developed to handle the spectrum assignment and handoff for
the SU traffic with two priority classes. We have developed the
analytical models for two proposed DSA policies. For perfor-
mance evaluation, we have derived the blocking probability,
the forced termination probability, and the throughput for the
two priority classes of SU traffic. We have also investigated the
case of sub-channel reservation for the high priority SUs and
obtained the optimal sub-channel reservation. The analytical
results have been verified through simulations.
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