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Optimal Service Rate in Cognitive Radio Networks With Different Queue Length Information

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Cognitive radio (CR) technology effectively overcomes spectrum inefficiency by providing the capability that unlicensed users can share the radio frequency spectrum with licensed users. In this paper, we consider a CR system with heterogeneous users. A single primary user (PU) randomly generates service requests and a licensed band processes these requests. This PU band also can be flexibly accessed by secondary users (SUs) if available. The CR system is regarded as preemptive priority queueing system. Under different information levels, we investigate the equilibrium strategic behaviors of PU and SUs. Based on users’ strategies, we study SU’s sojourn time (i.e., the period beginning from the time an SU request enters the system and ending from the time the SU request is completed), and obtain the analytical solutions of SU’s mean sojourn time. By theoretical and numerical analysis, the SU’s mean sojourn time is found not decreasing with the service rate of PU. This phenomenon is counterintuitive, and it implies the increase of the service rate of PU does not necessarily reduce the mean sojourn time of SU. In this sense, we investigate and find optimal service rates of PU in different information levels to meet the PU’s QoS requirement and simultaneously to maximize SU’s throughput from the viewpoint of the service providers. OAPA
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Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
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Optimal Service Rate in Cognitive Radio
Networks with Different Queue Length
Information
SHENG ZHU1,2, JINTING WANG1, and WEI WAYNE LI3, (Senior Member, IEEE)
1Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China (e-mail: jtwang@bjtu.edu.cn)
2School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454003, China (e-mail: shengzhu@hpu.edu.cn)
3Department of Computer Science and the NSF Center for Research on Complex Networks, Texas Southern University, Houston, TX 77004, USA (e-mail:
liww@tsu.edu)
Corresponding author: Jinting Wang (e-mail: jtwang@bjtu.edu.cn).
This work was supported in part by the National Natural Science Foundation of China under Grant no. 71571014 and no. 71871008, the
111 Project of China (B16002), and US National Science Foundation under Grant no. 1137732.
ABSTRACT
Cognitive radio (CR) technology effectively overcomes spectrum inefficiency by providing the capability
that unlicensed users can share the radio frequency spectrum with licensed users. In this paper, we
consider a CR system with heterogeneous users. A single primary user (PU) randomly generates service
requests and a licensed band processes these requests. This PU band also can be flexibly accessed by
secondary users (SUs) if available. The CR system is regarded as preemptive priority queueing system.
Under different information levels, we investigate the equilibrium strategic behaviors of PU and SUs.
Based on users’ strategies, we study SU’s sojourn time (i.e., the period beginning from the time an SU
request enters the system and ending from the time the SU request is completed), and obtain the analytical
solutions of SU’s mean sojourn time. By theoretical and numerical analysis, the SU’s mean sojourn time
is found not decreasing with the service rate of PU. This phenomenon is counterintuitive, and it implies
the increase of the service rate of PU does not necessarily reduce the mean sojourn time of SU. In this
sense, we investigate and find optimal service rates of PU in different information levels to meet the PU’s
QoS requirement and simultaneously to maximize SU’s throughput from the viewpoint of the service
providers.
INDEX TERMS Cognitive radio (CR), Nash equilibrium, quality of service (QoS), queueing system,
sojourn time, throughput.
I. INTRODUCTION
RADIO spectrum is an important scarce resource for
wireless communications. With the rapid increase of
data-processing business, the scarcity of spectrum becomes
a serious problem and the traditional static spectrum allo-
cation policy faces a difficult situation [1]. To solve this
problem, a dynamic spectrum access technology named
cognitive radio (CR) was first introduced by Mitola [15].
CR technology effectively solves the spectrum inefficiency
problems by providing unlicensed users the capability to
share the radio frequency spectrum with licensed users. It
has been rapidly developed and is widely used in different
engineering fields.
In general, users are heterogeneous and can be divided
into two types, namely, licensed users and unlicensed users.
The spectrum is licensed to the former while the latter also
can use the spectrum to transmit their packets if a spectrum
band is idle. Licensed users have priorities over unlicensed
users. When an arriving licensed user finds the band is
occupied by an unlicensed user, he will eject the serving
unlicensed user and occupy the band. Users of the same
type are served according to the order of arrivals. Based
on the description above, the cognitive radio system can be
modeled as a queueing system.
For the cognitive radio networks with two types of users,
Do et al. [6] considered the CR system as a queueing
system with breakdowns, where arrival process of service
requests is Poisson process and the time each user occupies
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Sheng Zhu et al.: Optimal Service Rate in Cognitive Radio Networks with Different Queue Length Information
the spectrum (i.e., service time) is exponentially distributed.
Subsequently, the model was extended to the case that the
service time is not exponential [7]. In reality, licensed user
has higher priority to access the band over unlicensed users,
so the queueing system based on CR networks framework
can be designed with preemptive priority queue [26]. Wang
and Li [22] considered the case that an arriving unlicensed
user retries to enter the system with a certain rate if the
band is occupied upon arrival. In their model, both licensed
user and unlicensed users are not permitted to wait before
the spectrum, i.e., no buffer in the CR networks. A retrial
queueing system was used to model the CR networks. Fur-
ther, when the spectrum sensing is unreliable, the system can
be modeled as the retrial queueing system with breakdowns
and recoveries [21]. As we know, the throughput of the
system with buffers is greater than that without buffer. In
this paper, we considered the CR networks as preemptive
priority queueing system with heterogeneous users and two
infinite buffers.
A type of opportunistic spectrum access called decentral-
ized cognitive medium access control protocol is studied in
this paper. With this protocol, licensed users and unlicensed
users can autonomously make decisions on spectrum access.
Users are selfish and their objectives are to maximize
their own net benefit. Although users’ strategic behavior
has crucial impact on the performance of CR networks,
many works in literatures ignored users’ strategies when
discussing the system performance and the optimal decision
of the managers [5]. Recently, game-theoretic studies have
been paid much attention to explore users’ equilibrium
strategic behavior. Interested readers can refer to [4], [8],
[9], [11], [12], [17] [23], [24] and [27] for more details on
this subject. For the CR networks, users’ strategic behaviors
have also been concerned and the relevant works include [2],
[6], [7], [13], [16], [18], [19] and [20]–[22], among others.
From the viewpoint of the managers of the CR net-
works, quality of service (QoS) and throughput are two
important concerns. Ma et al. [14] considered an adaptive
power allocation with the quality guarantee of service in
the cognitive radio networks. Amer et al. [3] studied the
maximum throughput of CR network with energy harvesting
in the case that the two types of users (i.e., licensed users
and unlicensed users) are cooperative.
In this paper, all licensed users are considered to be
symmetric. Therefore, we only study the case of a single
licensed user who randomly sends service requests. Accord-
ing to the IEEE 802.11 protocol, an arriving unlicensed user
will enter a virtual waiting space if the PU band is unavail-
able, and his packet is immediately transmitted if idle. In the
rest of the paper, licensed user, unlicensed user are called
primary user (PU) and secondary user (SU), respectively.
The performance analysis of the system and the optimal
decision of the manager are presented. We first summarize
users’ strategic behaviors, and then the performance of the
CR networks is discussed based on users’ decisions. We
maximize the throughput of SU requests under the condition
that the QoS meets the requirement of PU, and then obtain
the optimal service rates of PU in different information
cases. Wang and Li [22] is closely relevant to this paper.
In [22], the CR networks are assumed to have no buffer
for PU requests. They used a retrial queue to model the
CR networks, and obtained individual optimal strategy and
socially optimal strategy in cognitive radio networks. In
contrast to [22], we consider the CR networks with two
types of buffers in this paper. The existence of buffers will
improve the throughput of the system. We also consider the
quality of service (QoS), and obtain optimal decision of the
managers under the condition that the QoS is satisfied.
In a recent work [25], we studied the users’ equilibrium
joining strategies in cognitive radio network from the view-
point of different types of users. Based on three different
information levels, Wang et al. [25] obtained users’ strategic
behavior including Nash balance PU (SU) threshold policy
and equilibrium joining probability, and found SUs’ equilib-
rium joining probability does not necessarily increase with
the transmission rate of PU. In contrast to [25], we study
the performance measure of the system and optimal decision
from the viewpoint of the service provider (SP) in this paper.
In summary, the main contributions of this paper are:
The formulas of SU’s mean sojourn time are derived.
Through theoretical and numerical analysis, we find
a counterintuitive phenomenon that the SU’s mean
sojourn time does not necessarily decrease with the
service rate of PU. It deviates from our intuition.
Specifically, the sojourn time of an arriving SU is non-
decreasing first and non-increasing subsequently in no
queue length information case, and it presents more
variability in partial and full queue length information
cases. So the increase of the service rate of PU does
not necessarily reduce the mean sojourn time of SU.
The quality of service (QoS) and the throughput (i.e.,
the efficient arrival rate of requests) are two important
concerns for the manager of CR network. Under the
QoS constraint, we obtain feasible intervals of the
service rate of PU, and then investigate the optimal
service rates of PU in different information levels to
meet the PU’s QoS requirement and simultaneously
to maximize SU’s throughput from the viewpoint of
the service provider.
This paper is organized as follows. Section II presents
model description and gives some notations. In Section III,
we show users’ strategic behaviors in different informa-
tion structures. Based on users’ decisions, we consider the
performance of the CR networks in Section IV. Section
V obtains the optimal decision from the viewpoint of
the managers under the condition that the QoS meets the
requirement of PU. Finally, some conclusions are given in
Section VI.
II. MODEL AND NOTATION
In this section, we provide our model and some notations
used in this paper. we consider a cognitive radio system with
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Sheng Zhu et al.: Optimal Service Rate in Cognitive Radio Networks with Different Queue Length Information
a single PU who randomly sends service requests. PU has
high priority to get service, and SUs can be served only
when there is no PU request in the system. The arrival
processes of PU requests and SUs are assumed to be Poisson
processes and the corresponding Poisson intensities are α
and λ, respectively. Each SU carries one request for service.
The PU band can transmit either one SU packet or one PU
packet at a time. The time each packet needs to occupy
the PU band is called the service time. Assume the service
time of PU request is exponential with rate βand the service
time of SU also follows exponential distribution with rate
µ.βand µare called the service rates of PU and SU,
respectively. Compared with SUs, PU has the priority to
be served. If an arriving PU finds the PU band is occupied
by another PU request, the arriving request will line up in
the PU-buffer; otherwise, it will occupy the PU band no
matter whether the PU band is idle or occupied by an SU
request. That means, if an arriving PU finds the PU band
is occupied by an SU request, the PU request will eject the
SU request and occupy the PU band. After all PU requests
in the system have been completed, the PU band resumes to
transmit the interrupted SU request. For an arriving SU, if
it finds the band is occupied by other users, he will line
up in the SU-buffer. The same priority users are served
according to the order of arrivals (i.e., first come fist service
principle). The cognitive radio system can be described as
a preemptive priority queueing system with heterogenous
users and a single server, where the PU band is considered
as the server.
Some necessary assumptions and notations are given as
follows. RP U denotes the reward received by PU after a PU
request is completely served, which reflects the satisfaction
of PU. RSU is the reward to the served SU for serving an SU
request. Let f1(x), f2(x)be the waiting costs of PU and SU
per average waiting xtime units, respectively. The waiting
cost functions can be assumed to be positive, increasing,
continuous functions (see Guo and Zipkin [10]) and their
inverse functions are defined as f1
1(x), f 1
2(x). Denote by
ρ1=α/β, ρ2=λ/µ the traffic intensities of PU requests
and SU requests, respectively.
We assume the arrival rates and the service rates are open
to all users. However, in reality, the information about the
queue length in the PU-buffer and the queue length in the
SU-buffer is not always open to every users. According to
the information level about the queue lengths, we consider
three scenarios including no queue length information case,
partial queue length information case, and full queue length
information case, which are defined as follows (see Table
1). Based on different information levels, we will investigate
and find optimal service rates of PU in different information
levels to meet the PU’s QoS requirement and simultaneously
to maximize SU’s throughput (i.e., the effective arrival rate
of SU requests) from the viewpoint of the service providers
(see Section V), namely
max
βnλeff
SU,i(β)
PU’s QoS is satisfactoryo,(1)
TABLE 1. Three Information Levels Present in This Paper
Information level Explanation
No queue length in-
formation case
Both PU and SUs does not get any informa-
tion about the queue lengths of PU requests
and SU requests in the system.
Partial queue length
information case
PU only knows the queue length of PU
requests in the system and SUs only know
the queue length of SU requests in the
system.
Full queue length in-
formation case
Both PU and SUs can obtain all queue
length information including the number of
PU requests in the system and the number
of SU requests in the system.
where λeff
SU,i(β), i = 0,1,2, are the throughputs of SU
requests in no, partial and full queue length information
cases, respectively.
III. NASH EQUILIBRIUM STRATEGIES
In this paper, our objective is to explore the optimal decision
from the viewpoint of the CR managers. However, users’
joining strategy has important impacts on the performance
of the system. Therefore, before discussing the optimal
decision, we need to consider users’ equilibrium strategy.
All users are assumed to be selfish, so we consider
users are strategic. Upon arrival, they decide their joining
strategies from the viewpoint of maximizing their own net
benefit. There are three joining strategies to be considered,
namely, joining the system, balking and a mixed strategy
(i.e., joining the system with a certain probability). An
arriving PU (or SU) request will join the system if joining
makes the PU (or SU) gain the positive expected net benefit,
balk if negative, and is indifferent between joining and
balking if the expected net benefit equals zero. The expected
net benefit is the reward for receiving service minus the
expected waiting cost.
The expected net benefit of an arriving user is dependent
of the information available to an arriving user, since his
expected waiting cost is estimated based on the information.
Therefore, users will adopt different joining strategies under
different information levels.
According to the work of Wang, Zhu and Li [25], users’
equilibrium joining strategies in three different information
levels are given as follows.
In the no queue length information case, there exists
equilibrium strategy (qe
P U ,qe
SU )such that “PU requests
join (i.e., are send to) the system with probability qe
P U and
SU requests join the system with probability qe
SU ” is an
equilibrium, where
qe
P U = min nβ
α1
αf1
1(RP U ),1o,(2)
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Sheng Zhu et al.: Optimal Service Rate in Cognitive Radio Networks with Different Queue Length Information
and
qe
SU =
0RSU f2(WS U (0)) ,
¯q f2(WSU (0)) < RSU < f2(WSU (1)),
1RSU f2(WS U (1)),
(3)
in which ¯qis the solution to RS U f2(WSU (q)) = 0 and
WSU (q) = ρ2ρ1qe
P U ρ2+ρ1qe
P U ρ2(µ/β)
λ(1 ρ1qe
P U )(1 ρ1qe
P U ρ2q).(4)
Remark 3.1: If βα+ 1/f1
1(RP U ), from (2) we have
qe
P U = 1. Therefore, in this situation, each arriving PU
request will join the system.
In the partial queue length information case, there exists
threshold strategy (ne
P U , ne
SU ), such that “an arriving PU
(SU) request will join the system if the number of PU (SU)
requests in the system upon arrival is less than and equals
ne
P U (ne
SU ) and otherwise balk.” is an equilibrium, where
ne
P U =bβf1
1(RP U )1c,(5)
and
ne
SU =max njµ(f1
2(RSU )
ne
P U+1
P
i=1
PP U (i)¯
Bi)
1+α¯
B1k1,0
o,(6)
in which
¯
Bj=j
β(1ρ1)ρne
P Uj+2
1(1ρj
1)
β(1ρ1)2, j = 1,2,· · ·,ne
P U+1,(7)
and
PP U (i) = (1 ρ1)ρi
1
1ρne
P U +2
1
, i = 0,1,2,·, ne
P U + 1.(8)
Actually, ¯
Bjin (7) is the expected busy period induced by j
PUs. Interested readers can refer to [28] and [25] for more
details about the busy period.
In the full queue length information case, there exists
threshold strategy (ne
P U , ne
SU (0), ne
SU (1),· · · , ne
SU (ne
P U +
1)), such that the strategy “an arriving PU request will join
the system if the number of PU requests in the system upon
arrival is less than and equals ne
P U and balk otherwise. In
addition, an arriving SU observes the number of PUs in the
system, k, and the number of SUs in the system, N; he will
join the system if Nne
SU (k)and otherwise balk” is an
equilibrium, where ne
P U can be obtained from (5) and
ne
SU (k) = max µ(f1
2(RSU )¯
Bk)
1 + α¯
B11,0,(9)
and ne
P U ,¯
Bkare given in (5) and (7), respectively.
IV. SOJOURN TIME
In this section, we discuss the performance of the cognitive
radio network. As a key performance measure, the mean
sojourn time is considered. In the real situation, to guarantee
the QoS meets the requirement of PU, the mean sojourn
time of PU request needs to be limited in a certain interval
which will be explored in Section V. Here we ignore the
analysis about the mean sojourn time of PU request and
only consider the mean sojourn time of SU. We will find
the increase of the service rate of PU does not necessarily
reduce the mean sojourn time of SU.
In the rest of the paper, “PU (or SU) adopts strategy q
means an arriving PU (or SU) request joins the system with
probability q, and “PU (or SU) adopts the threshold strategy
n” denotes an arriving PU (or SU) request joins the system
if the number of PU (or SU) requests in the system is not
greater than nupon arrival.
A. NO QUEUE LENGTH INFORMATION CASE
We first consider that the mean sojourn time of an arriving
SU in no queue length information case. Assume the steady
condition of the system holds, i.e., ρ1+ρ2<1. After con-
ducting some algebra, the steady condition can be written
as
β > µα
µλ.(10)
Under the condition that PU adopts ‘strategy qe
P U ’ and all
other SUs adopt ‘strategy q’, the monotonicity of the mean
sojourn time on the service rate of PU is given in Proposition
4.1.
Proposition 4.1: Under the condition that PU adopts ‘s-
trategy qe
P U ’ and all other SUs adopt ‘strategy q’, the
mean sojourn time of an arriving SU increases with βin
the interval (µα/(µλ), α + 1/f1
1(RP U )) if µλ
λαf1
1(RP U )>0, and decreases with βin the interval
[max{α+ 1/f1
1(RP U ), µα/(µλ)},+).
Proof: The equilibrium joining probability of PU requests,
qe
P U , is given in (2). Substituting (2) into (4), we get
WSU =
f1
1(RP U)ρ2(βµ+f1
1(RP U)βµ)
λ(1 f 1
1(RP U)ρ2
),β
α1
αf1
1(RP U)<1,
(β2αβ+αµ)ρ2
λ(αβ)(αβ+qβρ2),β
α1
αf1
1(RP U )1.
(11)
To explore the monotonicity of WSU with respect to β, the
first-order derivative of the mean sojourn time is computed
as follows:
dWSU
dβ=
Θ1
λ(1f1
1(RP U )ρ2)2,β
α1
αf1
1(RP U )<1,
αρ2Θ2
λ(αβ)2(αβ+qβρ2)2,β
α1
αf1
1(RP U )1,(12)
where Θ1=f1
1(RP U )ρ2(1 + f1
1(RP U )µ(1 2)) and
Θ2= (βα)(αβ2µ) + q(2βα)µρ2. Obviously,
α, β, λ, ρ2, q , f1
1(RP U )are positive. As stated in Section
IV, ρ1+ρ2<1. Then we easily get
12>0,(13)
λβ +µα < µβ. (14)
If β/α 1/(αf 1
1(RP U )) <1, i.e., β < α + 1/f1
1(RP U ),
from (13), dWSU /dβ > 0. From (10), we get the mean
sojourn time of an arriving SU increases with βin the
interval (µα/(µλ), α + 1/f 1
1(RP U )) if µα/(µλ)<
α+ 1/f1
1(RP U ), i.e., µλλαf 1
1(RP U )>0.
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Sheng Zhu et al.: Optimal Service Rate in Cognitive Radio Networks with Different Queue Length Information
Now we judge the monotonicity of WSU as β/α
1/(αf1
1(RP U )) 1. From (14), we get
Θ2=(βα)22µβ + 2µα + 2λ qαλ
<(βα)22(λβ +µα)+2µα + 2qβλ qαλ,
=(βα)22λβ(1 q)qαλ < 0.(15)
From (15), dWSU /dβ < 0if β1/(αf 1
1(RP U )) 1,
i.e., βα+ 1/f1
1(RP U ). Therefore, the mean sojourn
time of an arriving SU decreases with βin the interval
[max{α+ 1/f1
1(RP U ), µα/(µλ)},+).
Remark 4.2: Under the condition that PU adopts ‘strategy
qe
P U ’ and all other SUs adopt ‘strategy q’, the mean sojourn
time of an arriving SU first increases and then decreases
with βif µλλαf1
1(RP U )>0.
Remark 4.2 can be easily obtained from Proposition
4.1, since µλλαf1
1(RP U )>0is identical to
α+ 1/f1
1(RP U )> µα/(µλ). The phenomenon showed
in Remark 4.2 is counterintuitive. In our intuition, the mean
sojourn time of an arriving SU decreases with the service
rate of PU. The reason is given as follows. As stated in
Section II, PU is higher priority user than SUs, and PU
requests can preempt SU request in service. As the service
rate of PU grows, the mean service time that each PU
request needs to spend is decreasing. It seems that the
tagged SU request can obtain service in a short time. And
then it seems reasonable that the mean sojourn time of an
arrival SU decreases with the service rate of PU, because
PU requests in the system can be emptied with fast rate.
However, our intuition is wrong. Proposition 4.1 shows the
mean sojourn time of an arrival SU increases first and then
decreases with βif µλλαf1
1(RP U )>0. As the service
rate of PU increases, more PU requests will be sent to the
system. The tagged SU request needs to spend longer time
waiting in the system due to more congested system and
more preemptions during the service period of the tagged
SU request. From (2), if the service rate of PU, β, is greater
than a critical value α+ 1/f1
1(RP U ),qe
P U = 1, i.e., each
arriving PU request always joins the system. So the effective
arrival rate of PU request will not vary with increase of β
as βα+ 1/f1
1(RP U ). But PU requests in the system
will be served with faster rate as βgrows. Therefore, in this
situation, the mean sojourn time of an arrival SU decreases
with the service rate of PU. A numerical example is used
to verify Proposition 4.1. Fig. 1 shows that the relation
between the mean sojourn time of an arriving SU and the
service rate of PU. We find that the mean sojourn time of
an arrival SU increases first and then decreases with βfor
α= 0.4, λ = 0.2, µ = 1, q = 1, f 1
1(RP U )=3.
Remark 4.3: Under the condition that PU adopts ‘strate-
gy qe
P U ’ and all other SUs adopt ‘strategy q’, the mean
sojourn time of an arriving SU is weakly unimodal if
µλλαf1
1(RP U )>0. It can be directly obtained
from Proposition 4.1, and can be also observed in Fig. 1.
Remark 4.4: Under the condition that PU adopts ‘strategy
qe
P U ’ and all other SUs adopt ‘strategy q’, the mean sojourn
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
2
3
4
5
6
7
8
9
10
11
β
WSU
FIGURE 1. The mean sojourn time of an arriving SU vs. the service rate of
PU for α= 0.4, λ = 0.2, µ = 1, q = 1, f 1
1(RP U )=3.
time of an arriving SU decreases with βif
µλλαf1
1(RP U )<0.(16)
As stated earlier in this subsection, we assume that the
steady condition is satisfied, so β > µα/(µλ). If (16)
holds, we have
α+1
f1
1(RP U )<µα
µλ< β. (17)
According to Proposition 4.1, we immediately obtain Re-
mark 4.4. From Remark 3.1, all arriving PU requests will
join the system if (16) holds. With the increase of the service
rate of PU, the effective arrival rate of PU requests keeps
invariable, and PU requests in the system will be served
with faster rate. That means, the server will complete the
services for users lining before an arriving SU in a shorter
time, Therefore, in this situation, the mean sojourn time of
an arrival SU decreases with the service rate of PU.
Remark 4.5: In the equilibrium state, the mean sojourn time
of an arriving SU is nondecreasing first and non-increasing
subsequently in βif µλλαf1
1(RP U )>0. The
result implies that increasing the service rate of PU does
not necessarily reduce the mean sojourn time of SU.
In the equilibrium state, PU adopts ‘strategy qe
P U ’ and
SUs adopt ‘strategy qe
SU ’. Replacing qin Proposition 4.1
with qe
SU , we can obtain the result of Remark 4.5. Let
TSU be the mean sojourn time of an arriving SU in the
equilibrium state. A numerical example is given. Observing
Fig. 2, we easily find the phenomenon presented in Remark
4.5, and also find the mean sojourn time of an arriving SU
is weakly unimodel (see Remark 4.3).
B. PARTIAL QUEUE LENGTH INFORMATION CASE
In this subsection, we consider the mean sojourn time of
an arriving SU in partial queue length information case. Let
Tp(i)be the mean sojourn time of an arriving SU if he
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Sheng Zhu et al.: Optimal Service Rate in Cognitive Radio Networks with Different Queue Length Information
0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
β
TSU
f2
−1(RSU)=1.5
f2
−1(RSU)=2.5
FIGURE 2. The mean sojourn time of an arriving SU in the equilibrium state
vs. the service rate of PU for α= 0.4, λ = 0.2, µ = 1.
finds iSUs in the system upon arrival. According to [25],
the mean sojourn time can be computed from
Tp(i) = (i+ 1)(1 + α¯
B1)
µ+
ne
P U +1
X
j=1
PP U (j)¯
Bj,(18)
where ¯
Bjis determined by (7).
Proposition 4.6: If the threshold adopted by PU does not
change with βfor β[a, b),Tp(i)decreases with βin the
interval [a, b).
Proof: According to the condition of Proposition 4.6, the
threshold adopted by PU does not change with β[a, b).
Without loss of generality, we assume the threshold is a
constant value ˜ne
P U .Tp(i)is the mean sojourn time of
the arriving SU finding iSUs in the system. As βgrows,
the mean service time each PU request needs to spend is
decreasing. A tagged SU who finds iSU requests in the
system upon arrival can obtain service in a short time. In
addition, with increase of β(β[a, b)), the effective arrival
rate of PU requests keeps invariable since the threshold
adopted by PU does not change with βas β[a, b).
Therefore, the increase of βmakes the server serve PU
requests with faster rate, but does not induce more PU
requests to join the system. Then the mean sojourn time
of the tagged SU decreases with β. We obtain Proposition
4.6.
Proposition 4.7: Assume the threshold adopted by PU does
not change with βas β[a, b)or β[b, c). If the
threshold corresponding to β(a, b]is different from that
corresponding to β[b, c),Tp(i)occurs a jump on β=b.
Proof: We still assume the threshold adopted by PU is
˜ne
P U as β[a, b). From (5), we easily find the threshold
adopted by PU is non-decreasing in β. According to the
condition of Proposition 4.7, the threshold corresponding to
β(a, b]is different from that corresponding to β[b, c),
so the latter equals ˜ne
P U +1.Tp(i)is dependent of β. For the
convenience of explanation, we redefine Tp(i)as Tp(i, β).
When an arriving SU finds iSUs in the system, the mean
sojourn time of the arrival is Tp(i, b)if βband
0.6 0.7 0.8 0.9 1 1.1 1.2
4.5
5
5.5
6
6.5
7
7.5
β
Tp(2)
β=0.67
β=1
FIGURE 3. Tp(2) vs. βfor α= 0.4, λ = 0.2 = 1, f1
1(RP U )=3.
Tp(i, b+) if βb+. PU adopts the threshold strategy ˜ne
P U
as βband ˜ne
P U + 1 as βb+. When βvaries from
bto b+, more PU requests are permitted to join the system
since the threshold strategy adopted by PU varies from ˜ne
P U
to ˜ne
P U + 1. Therefore, the arriving SU needs to spend more
time sojourning in the system due to more congested system
and more preemptions. Then Tp(i, b+) > Tp(i, b), and
Proposition 4.7 is obtained.
A numerical example is used to explore the feature of
the sojourn time. Fig. 3 shows the tendency of Tp(2) with
increase of βfor α= 0.4, λ = 0.2, µ = 1, f1
1(RP U ) = 3.
From (5), we easily obtain the threshold adopted by PU as
follows: ne
P U = 0 if β[0.6,0.67),ne
P U = 1 if β
[0.67,1), and ne
P U = 2 if β[1,1.2]. Observing Fig. 3,
we find Tp(2) decreases with βin [0.6,0.67),[0.67,1) or
[1,1.2]. But Tp(2) has jumps as βis on 0.67 and 1.
C. FULL QUEUE LENGTH INFORMATION CASE
In this subsection, we consider the mean sojourn time of
an arriving SU in full queue length information case. Let
Tf(i, k)be the mean sojourn time of an arriving SU if he
finds iSUs and kPU requests in the system upon arrival.
According to [25], it can be computed from
Tf(i, k) = (i+ 1)(1 + α¯
B1)
µ+¯
Bk.
Adopting the analysis method same to Proposition 4.6 and
Proposition 4.7, we can obtain similar results in the full
queue length information case.
Proposition 4.8: If the threshold adopted by PU does not
change with βfor β[a, b),Tf(i, k)decreases with β
[a, b).
Proposition 4.9: Assume the threshold adopted by PU does
not change with βas β[a, b)or β[b, c). If the
threshold corresponding to β[a, b)is different from that
corresponding to β[b, c),Tf(i, k)occurs a jump on
β=b.
We ignore the proofs of Proposition 4.8 and Proposition
4.9 since the proofs are similar to Proposition 4.6 and Propo-
sition 4.7 respectively. Fig. 4 shows the relation between
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Sheng Zhu et al.: Optimal Service Rate in Cognitive Radio Networks with Different Queue Length Information
0.6 0.7 0.8 0.9 1 1.1 1.2
5.5
6
6.5
7
7.5
8
8.5
β
Tf(2,1)
β=0.67
β=1
FIGURE 4. Tf(2,1) vs. βfor α= 0.4, λ = 0.2 = 1, f1
1(RP U )=3.
Tf(2,1) and βfor α= 0.4, λ = 0.2, µ = 1, f 1
1(RP U ) =
3. We find Tf(2,1) decreases with βin [0.6,0.67),[0.67,1)
or [1,1.2), and Tf(2,1) has jumps as βis on 0.67 and 1.
Through the analysis above, it can be found that the mean
sojourn time of each arriving SU often does not decrease
with the service rate of PU, so increasing the service rate
of PU does not necessarily reduce the mean sojourn time of
SU.
V. OPTIMAL SERVICE RATE
The quality of service (QoS) is an important issue deserving
to be considered by the manager of CR network. As primary
users, the QoS to PU always is preferentially guaranteed.
In addition, the size of the throughput of SU requests
embodies the efficiency of CR networks. In this section, we
consider the optimal service rate of PU which maximizes
the throughput of SU requests, under the condition that the
QoS to PU is satisfactory.
Let We
P U,0(β)be the mean sojourn time of joining PU
request in no queue length information case, We
P U,1(β)is
the mean sojourn time of joining PU request in partial queue
length information case, and We
P U,2(β)denotes the mean
sojourn time of joining PU request in full queue length
information case. We also assume that β
0,β
1, and β
2are
the optimal service rates of PU in three information cases,
respectively. In reality, to guarantee the QoS to PU, the
service provider provides an appropriate service rate for PU
such that the mean sojourn time of PU request is not greater
than a given value. In addition, too short mean sojourn
time will induce high technique cost. So it is reasonable
to assume the mean sojourn time of PU request is greater
than and equals a lower bound. Hence there exist νlow and
νup such that the mean sojourn time of PU request satisfies
νlow We
P U,i(β)νup , i = 0,1,2.(19)
According to the equation above, we can get the value range
of the service rate as follows:
{β:νlow We
P U,i(β)νup }, i = 0,1,2.(20)
As defined in Section II, λeff
SU,i(β), i = 0,1,2are the
throughputs of SU requests (i.e., the effective arrival rates
of SU requests) in three different information cases, re-
spectively. Now we find an optimal service rate of PU
to meet the PU’s QoS requirement and simultaneously to
maximize SU’s throughput from the viewpoint of the service
providers. The proposed problem is an optimal decision
problem and the corresponding mathematical expression is
given as follows:
max λeff
SU,i(β)(21)
subject to νlow We
P U,i(β)νup ,(22)
where i= 0,1,2. If the above optimization problem has
multiple optimal solutions, the smallest one is the optimal
service rate of PU since the increase of the service rate
needs to spend cost.
Definition 5.1: The intervals i={β:νlow We
P U,i(β)
νup}, i = 0,1,2,are called feasible intervals in the corre-
sponding information cases.
According to the optimization problem (21)-(22) and
Definition 5.1, the optimal service rate of PU can be written
as
β
i= min arg max
βi
λeff
SU,i(β), i = 0,1,2.(23)
A. NO QUEUE LENGTH INFORMATION CASE
In the no queue length information case, PU can not obtain
the queue length information of PU requests. According to
(2), an arriving PU request joins the system with equilibrium
probability qe
P U . So PU requests effectively arrive in the
system with rate λqe
P U and are served with rate β. By
using the primary result of M/M/1queue system, the
mean sojourn time of joining PU request can be obtained
as follows:
We
P U,0(β) = 1
βαqe
P U
.(24)
Lemma 5.2: In the no queue length information case, the
feasible interval can be written as
0=
, f1
1(RP U )< νlow ,
0, α+1
νlow , νlow f1
1(RP U )νup,
α+1
νup , α+1
νlow , f1
1(RP U )> νup .
(25)
Proof: From (2), the equilibrium joining probability of PU
requests, qe
P U , can be rewritten as
qe
P U =
1, β α+1
f1
1(RP U ),
β
α1
αf1
1(RP U ), β < α +1
f1
1(RP U ).(26)
By (24) and (26), the mean sojourn time of joining PU
request is
We
P U,0(β) =
1
βα, β α+1
f1
1(RP U ),
f1
1(RP U ), β < α +1
f1
1(RP U ).(27)
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
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Sheng Zhu et al.: Optimal Service Rate in Cognitive Radio Networks with Different Queue Length Information
0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
β
λeff
SU,0
β=1.5
β=1.4
(1.5,0.104)
FIGURE 5. The throughput of SU requests λef f
SU,0(β)in the no queue length
information case vs the service rate βfor α= 0.4, λ = 0.2, µ = 1,
f1
1(RP U )=1.5, f 1
2(RSU )=2, νlow = 0.9, νup = 1.
In the no queue length information case, the feasible interval
0={β:νlow We
P U,0(β)νup}. From (27), 0can
be rewritten as
0={β|β < α+1
f1
1(RP U )and νlow f1
1(RP U )νup}
∪{β|βα+1
f1
1(RP U )and νlow 1
βανup}.(28)
After some computations, we get (25). This completes the
proof.
In the no queue length information case, the effective
arrival rate of SU requests is the potential total arrival rate
times the equilibrium joining probability, namely
λeff
SU,0(β) = λqe
SU .(29)
(29) can be easily computed since λis a given value and qe
SU
can be obtained from (3). The effective arrival rate of SU
requests in the no queue length information case λeff
SU,0(β)
is a function of βsince qe
SU depends on WS U (q)which is
a function of β(see (3) and (4)). According to (23) and
(29), the optimal service rate of PU in the no queue length
information case is given by
β
0= min arg max
β0
λqe
SU .(30)
Fig. 5 shows the relation between the throughput of SU
requests and βin the no queue length information case.
According to Lemma 5.2, we can obtain the feasible interval
0= [1.4,1.5] as α= 0.4, λ = 0.2, µ = 1, f1
1(RP U ) =
1.5, f 1
2(RSU ) = 2, νlow = 0.9, νup = 1. Therefore, to
maximize the throughput of SU requests, we need to search
the optimal service rate β
0in the feasible interval [1.4,1.5].
Observing Fig. 5, we find the throughput of SU requests
is maximized as the service rate β= 1.5, so the optimal
service rate β
0= 1.5.
B. PARTIAL QUEUE LENGTH INFORMATION CASE
In the partial queue length information case, PU can obtain
the queue length information of PU requests. As stated
0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
β
ne
SU
(0.92,1)
β=0.84 β=1.1
1.1
0.84
FIGURE 6. The threshold ne
SU vs the service rate of PU βin the partial
queue length information for
α= 0.4, λ = 0.2, µ = 1, f1
1(RP U )=3, f 1
2(RSU )=4.
earlier in Section III, there exists an threshold ne
P U , such
that an arriving PU request will join the system if the
number of PU requests in the system is less than and equals
ne
P U upon arrival and otherwise balk. By using the primary
result of queue with truncation, we get
We
P U,i(β)=1(2+ne
P U )ρ1+ne
P U
1+(1+ne
P U )ρ2+ne
P U
1
β(1ρ1)(1ρ1+ne
P U
1),(31)
where i= 1,2. The analytical solutions of both 1and
2are difficult to be obtained. Some numerical analysis
can help us find the feasible intervals in the partial and full
queue length information cases.
For the partial queue length information case, the through-
put of SU requests, λeff
SU,1(β), has the same monotonicity
with the threshold adopted by PU, ne
SU . Then we can get
the optimal service rate of PU from
β
1= min arg max
β1
ne
SU ,(32)
where ne
SU is given by (6). Observing (6)-(8), we easily find
that the threshold ne
SU is the function of βand the exact
solution of the threshold can be obtained through simple
computations.
The sensitivity of the threshold ne
SU on βin the partial
queue length information is shown in Fig. 6. Given the
feasible interval 1= [0.84,1.1], from Fig. 6 we find in
the feasible interval 1there exist many different service
rates such that the threshold ne
SU is maximized. By (32), the
optimal service is the smallest value among these maximum
points. So we easily find the optimal service β
1= 0.92
given 1= [0.84,1.1].
C. FULL QUEUE LENGTH INFORMATION CASE
Now we consider the full queue length information case.
ne
SU (j)depends on β. For the convenience’s sake, we
relabel it as ne
SU (β , j), namely, ne
SU (j),ne
SU (β , j). There
are two cases needing to be considered. One is the case
that the threshold adopted by PU does not vary with β
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Sheng Zhu et al.: Optimal Service Rate in Cognitive Radio Networks with Different Queue Length Information
0.6 0.8 1 1.2 1.4 1.6 1.8 2
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
β
ne
SU(k)
ne
SU(0)
ne
SU(1)
ne
SU(2)
ne
SU(3)
ne
SU(4)
ne
SU(5)
ne
SU(6)
β=1.64
β=1.34
β=1.6
FIGURE 7. The threshold ne
SU (k)vs the service rate of PU βin the full
queue length information for
α= 0.4, λ = 0.2, f 1
1(RP U )=3, f 1
2(RSU )=8.
belonging to the feasible interval 2, and the other is the
case that the threshold may change as βvaries within the
feasible interval. For the second case, we can not obtain a
good formula about the optimal service rate. It is an open
problem and deserves further study. We only consider the
optimal service rate of PU in the first case. The threshold
adopted by PU is ne
P U , which is given in (5). An arriving
PU request will join the system if there are at most ne
P U
PU requests in the system upon arrival; otherwise balk. If
ne
SU (ˆ
β, j )ne
SU (β , j)for j= 0,1,2,· · · , ne
P U + 1,
obviously λeff
SU,2(ˆ
β)λeff
SU,2(β). Therefore from (23) we
can get the optimal service rate of PU as follows:
β
2= min nˆ
β|ne
SU (ˆ
β, j )ne
SU (β , j), f or β2,
and j= 0,1,2,· · · , ne
P U + 1o.
(33)
A numerical example is given here for α= 0.4, λ =
0.2, f 1
1(RP U ) = 3, f 1
2(RSU ) = 8. Given the feasible in-
terval 2= [1.34,1.64] in the full queue length information
case, we need to search the optimal rate β
2in this interval.
From (5), we get ne
P U = 3 as β2. Observing Fig. 7,
we find
ne
SU (ˆ
β, j )ne
SU (β , j),for β2,
and j= 0,1,2,· · · , ne
P U + 1,(34)
if ˆ
β[1.6,1.64]. From (33), we can obtain the optimal
service rate β
2= 1.6.
In summary, the optimal service rate of PU in the no
queue length information case can be computed from (30).
For the partial and full queue length information cases,
the corresponding optimal solutions can be determined
by (32) and (33), respectively. Through computing (30),
(32) and (33), the service provider can make the optimal
decision (i.e., set the optimal service rate), under which SUs’
throughput is maximized and simultaneously PU’s QoS is
satisfactory.
VI. CONCLUSIONS
In this paper we studied a CR system with two buffers
(PU request buffer and SU request buffer) and two classes
of users. According to the level of information disclosure,
we investigated three information cases, namely, no queue
length information (queue length information of both PU
requests and SU requests is concealed), partial queue length
information (queue length information is partial disclosed)
and full queue length information (queue length information
is fully open). Users’ equilibrium strategic behaviors in
different information cases have been summarized. The
strategies of PU and SUs have important impacts on the
performance of the CR networks. We explored the system
performance in equilibrium state and found that the sojourn
time of SU is non-increasing as the service rate of PU
grows, but the sojourn time of SU is nondecreasing first and
then non-increasing in no queue length information case. In
partial and full queue length information cases, the mean
sojourn time of SU has some jumps with the increase of the
service rate of PU. That means the increase of the service
rate does not necessarily improve the throughput of the CR
network. In addition, the QoS and the throughput are two
important concerns for the manager of CR network. We
take account of both two, and derive the optimal service
rates of PU under different information structures from the
perspective of the manager. For future work, one may extend
our model to the trilateral game among PUs, SUs and the
manager of CR network. Furthermore, based on the model
presented in this paper, the topics on energy saving and
energy harvest deserve to be explored.
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http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2018.2867049, IEEE Access
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SHENG ZHU received the B.Sc. degree from
Fuyang Normal University, Fuyang, China, in
2004, the M.Sc. degree from Chongqing Univer-
sity, Chongqing, China, in 2007, and now is a
Ph.D. candidate in Beijing Jiaotong University,
Beijing, China.
He is a lecturer in the School of Mathematics
and Information Science, Henan Polytechnic U-
niversity, Jiaozuo, China. He is a member of the
Operations Research Society of China (ORSC).
His research interests include queueing theory, the applications of game
theory and queueing theory in wireless communication and cloud com-
puting, financial mathematics and engineering. He have published more
than 10 papers in the proceedings of international conferences and interna-
tional professional journals such as Operational Research: An international
Journal, Journal of Industrial and Management Optimization, Advances in
Information Science and Service Sciences, etc.
JINTING WANG received the B.Sc. degree from
Hebei Normal University, Shijiazhuang, China, in
1994, the M.Sc. degree from Hebei University
of Technology, Tianjin, China, in 1997, and the
Ph.D. degree from the Chinese Academy of Sci-
ences, Beijing, China, in 2000.
He is a Professor and the Deputy director in
the Department of Mathematics, Beijing Jiaotong
University, Beijing, China. His research interests
include issues related to queueing theory, relia-
bility and the applications of game theory and queueing theory in wireless
communication and networking. He has published over 80 papers in
international journals such as IEEE Transactions on Vehicular Technol-
ogy, IEEE Transactions on Cognitive Communications and Networking,
Production and Operations Management, Queueing Systems, European
Journal of Operational Research, Journal of Multivariate Analysis, Journal
of Network and Computer Applications, etc. He is a member of the
Operations Research Society of China (ORSC), and now he serves as the
President of Reliability Society and the Vice-Presidents of the Queueing
Society affiliated with ORSC, and as Vice-President of Beijing Operations
Research Society. He was the recipient of the Outstanding Research Award
for Young Researchers from ORSC in 2004. In 2011, he was honored
with the Program for New Century Excellent Talents in University by the
Ministry of Education of China.
Dr. Wang is currently serving as an Editor for several professional
journals such as International Journal of Operations Research, International
Journal of Smart Grid and Green Communications and other two Chinese
journals.
WEI WAYNE LI (M’99-SM’06) received the
B.Sc.degree from Shaanxi Normal University,
Xi’an, China, in 1982, the M.Sc. degree from
Hebei University of Technology, Tianjin, China,
in 1987, and the Ph.D. degree from the Chinese
Academy of Sciences, Beijing, China, in 1994.
He is a Professor and the Director/PI of the
National Science Foundation (NSF) Center for
Research on Complex Networks, at the Texas
Southern University, Houston, TX, USA. He was
also once an Associate Professor with tenure in the Department of Electrical
Engineering and Computer Science, University of Toledo, Toledo, OH,
USA, and a tenure track Assistant Professor in the Department of Electrical
and Computer Engineering, University of Louisiana at Lafayette, LA,
USA. He is the author/co-author of 5 books and over 150 peer-reviewed
papers in professional journals and the proceedings of conferences, includ-
ing IEEE/ACM Transactions on Networking, IEEE Journal on Selected
Areas in Communications, IEEE Transactions on Communications, IEEE
Transactions on Wireless Communications, IEEE Transactions on Vehic-
ular Technology, Advances in Applied Probability, and INFOCOM et al.
His research interests include dynamic control, optimization, evaluation,
complexity, power connectivity, adaptation, design and implementation of
various advanced wireless systems.
Dr. Li is currently serving as an Editor for three professional journals, is
serving or has served as a Steering Committee Member/ General Co-Chair/
TPC Co-Chair/ Publicity Chair/ Session Chair/ TPC members, respectively,
for a number of professional conferences, such as INFOCOM, Globecom,
ICC, and WCNC et al.
10 VOLUME 4, 2016
... In this situation, a preemptive priority queueing system with two infinite buffers was adopted. Based on the queueing theory, a large amount of literatures focused on the network stability of CRNs [15], performance analysis [9,10], throughput [14,16], quality of service [17,18] and optimal pricing [5], etc. Recently, some works focused on the management of CRNs from the perspective of social welfare maximization [19]. ...
... However, in previous literatures (see [11][12][13][14]), it is generally assume that PU has preemptive priority over SUs, i.e., an arriving PU request can push out the SU in service and occupy the spectrum. The interrupted SU will resume service once the spectrum is available again. ...
Article
Cognitive radio (CR) networks with a single primary user (PU) and multiple secondary users (SUs) are often modeled as a priority queueing system, in which PU has higher priority over SUs. It is an interesting problem whether or not PU should preempt the SU in service. We study the optimal policy from two different perspectives, namely, throughput and social welfare of the system, based on users’ strategic behavior. We first study users’ equilibrium strategic behavior and derive two dimensional equilibrium joining strategies under a natural cost-reward structure, and then the throughput and the social welfare in the preemptive mechanism are compared with those corresponding to nonpreemptive case. It is surprising to find out that the nonpreemptive case is better than the preemptive case in some situations from the viewpoint of throughput maximization, but in other situations permitting preemptions is an optimal decision. While from the perspective of the social welfare, we observe that the nonpreemptive case is always better than the preemptive case. These results provide important managerial insights into how to design the service mechanism and control SUs in the CR systems.
... In [118], the authors have investigated equilibrium strategic behavior for two classes of users, i.e., Primary User (PU) and Secondary User (SU) based on three information cases, namely, no queue length information, partial queue length information, and full queue length information for CR systems. An efficient approach using theoretical and numerical analyses is derived to enhance PU and SU performance. ...
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With an extensive growth in user demand for high throughput, large capacity, and low latency, the ongoing deployment of Fifth-Generation (5G) systems is continuously exposing the inherent limitations of the system, as compared with its original premises. Such limitations are encouraging researchers worldwide to focus on next-generation 6G wireless systems, which are expected to address the constraints. To meet the above demands, future radio network architecture should be effectively designed to utilize its maximum radio spectrum capacity. It must simultaneously utilize various new techniques and technologies, such as Carrier Aggregation (CA), Cognitive Radio (CR), and small cell-based Heterogeneous Networks (HetNet), high-spectrum access (mmWave), and Massive Multiple-Input-Multiple-Output (M-MIMO), to achieve the desired results. However, the concurrent operations of these techniques in current 5G cellular networks create several spectrum management issues; thus, a comprehensive overview of these emerging technologies is presented in detail in this study. Then, the problems involved in the concurrent operations of various technologies for the spectrum management of the current 5G network are highlighted. The study aims to provide a detailed review of cooperative communication among all the techniques and potential problems associated with the spectrum management that has been addressed with the possible solutions proposed by the latest researches. Future research challenges are also discussed to highlight the necessary steps that can help achieve the desired objectives for designing 6G wireless networks.
... In [31], considering a dynamic spectrum access system in which SU could choose to either rent a licensed dedicated band or to use spectrum holes, the authors analyzed the equilibrium behavior for the SU decision strategies and applied the analysis results to maximize the revenue from renting dedicated bands to SUs based on the server-breakdown queueing model. In [32], the authors used the preemptive priority queueing model to investigate the different equilibrium strategic behaviors of SUs with no queue length information, partial queue length information and full queue length information. However, all the papers mentioned above assume that once the SU chooses to access the spectrum, it will not leave the system until the service transmission is completed. ...
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In the cognitive radio network (CRN), secondary users (SUs) compete for limited spectrum resources, so the spectrum access process of SUs can be regarded as a non-cooperative game. With enough artificial intelligence (AI), SUs can adopt certain spectrum access strategies through their learning ability, so as to improve their own benefit. Taking into account the impatience of the SUs with the waiting time to access the spectrum and the fact that the primary users (PUs) have preemptive priority to use the licensed spectrum in the CRN, this paper proposed the repairable queueing model with balking and reneging to investigate the spectrum access. Based on the utility function from an economic perspective, the relationship between the Nash equilibrium and the socially optimal spectrum access strategy of SUs was studied through the analysis of the system model. Then a reasonable spectrum pricing scheme was proposed to maximize the social benefits. Simulation results show that the proposed access mechanism can realize the consistency of Nash equilibrium strategy and social optimal strategy to maximize the benefits of the whole cognitive system.
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A vast body of literature usually has treated cognitive radio (CR) networks as single-band queues with preemptive priority for mathematical tractability, although, in practice, they are multiple-spectrum CR systems. In this article, we overcome these restrictions and employ a two-server preemptive priority queue to study CR systems with multiple spectrums. We obtain a joint optimal pricing strategy to maximize profit for the service provider (SP), as well as an optimal pricing strategy to maximize social welfare from the social administrator’s viewpoint. We find that the socially optimal pricing strategy is to provide free service for all PUs, whereas the joint optimal pricing strategy for SP’s maximum is equivalent to the pricing strategy that maximizes the two servers’ own expected net benefit.
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The current research efforts on Fifth Generation (5G) of wireless communication systems have identified the need for large extent improvements in accessibility and reliability of communication services. In this respect, Cognitive Radio (CR) has been envisioned as a key 5G enabler that allows dynamic spectrum access without causing interference to licensed (primary) users and can tackle the challenge of ultra reliable communication. Channel failures, which are generally caused due to hardware and software failure and/or due to intrinsic features such as fading and shadowing, can easily result in network performance degradation. In cognitive radio networks (CRNs), the connections of unlicensed (secondary) users are inherently vulnerable to breaks due to channel failures as well as licensed users’ arrivals. To explore the advantages of channel reservation and retrial phenomenon on performance improvement in error-prone channels, we propose and analyze dynamic spectrum access (DSA) scheme by also taking balking and reneging behavior into account. Moreover, since 5G networks should comprise heterogeneous applications that may have different Quality of Service (QoS), thus the present study facilitates the arrival of heterogeneous secondary users with access privilege variations. In addition, most previous works have studied the stationary performance of CRNs, however, those may not be adequate in practice, notably when the time horizon of operations is finite. This paper investigates the transient dynamics from the perspectives of dependability theory in CRNs. Furthermore, the whole system is modeled using a multi-dimensional continuous time Markov chain (CTMC) and numerical results illustrate the potential of the proposed scheme to achieve major gains in the performance of error-prone CRNs.
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In a cognitive radio (CR) system, excessive access services for secondary users (SUs) lead to a substantial increase in congestion and the retrial phenomenon, both of which degrade the performance of CR networks, especially in overload conditions. This paper investigates the price-based spectrum access control policy that characterizes the network operator’s provision to heterogeneous and delay-sensitive SUs through pricing strategies. Based on shared-use dynamic spectrum access (DSA), the SUs can occupy the dedicated spectrum without degrading the operations of primary users (PUs). The service to transmission of SUs can be interrupted by an arriving PU, while the interrupted SUs join a retrial pool called an orbit, later trying to use the spectrum to complete the service. In the retrial orbit, the interrupted SU competes fairly with other SUs in the orbit. Such a DSA mechanism is formulated as a retrial queue with service interruptions and general service times. Regarding the heterogeneity of delay-sensitive SUs, we consider two cases: the delay-sensitive parameter follows a discrete distribution and continuous distribution, respectively. In equilibrium, we find the revenue-optimal price is unique, while there may exist a continuum of equilibria for the socially optimal price. In addition, the socially optimal price is always not greater than the revenue-optimal price, and thus the socially optimal arrival rate is not less than the revenue-optimal one, which is contrary with the conclusion, i.e., the socially optimal and revenue-optimal arrival rates are consistent, drawn in literature for homogeneous SUs. Finally, we present numerical examples to show the effect of various parameters on the operator’s pricing strategies and SUs’ behavior.
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A cognitive radio (CR) system with retrial possibility and an admission cost for secondary users (SUs) to join the retrial group is investigated in this paper. If the SU finds the primary user (PU) band unavailable, it must decide with a probability estimate to either enter a retrial group or give up its service and leave the system. SUs in the retrial group independently retry after an exponentially distributed random time until they successfully access the spectrum. When the PU arrives, the SU's service on the band is interrupted. This interrupted SU is then assumed to occupy the PU band immediately when the PU completes its service. First, the noncooperative joining behavior of SUs that choose to maximize their benefit in a selfish distributed manner is investigated, and an inefficient Nash equilibrium is derived. Second, from the perspective of the social planner, the socially optimal joining strategy when SUs cooperate with each other is studied, and the corresponding Nash equilibrium is exactly derived. Finally, the result that an individually optimal strategy, in general, does not yield the socially optimal strategy is theoretically verified. Furthermore, to bridge the gap between the individually and socially optimal strategies, a novel strategy of imposing an admission fee on SUs to join the retrial group is proposed and investigated with the derivation of an optimal value for the admission fee. The numerical analysis indicates that the proposed admission fee as an equilibrium strategy and the socially optimal strategy of SUs improve efficiency in the utilization of the CR system.
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Multiplicity of solutions is typical for systems where the individual's tendency to act in a certain way increases when more of the other individuals in the population act in this way. We provide a detailed analysis of a queueing model in which two priority levels can be purchased. In particular, we compute all of the Nash equilibrium strategies (pure and mixed) of the threshold type.
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This paper deals with an N policy M/G/1 queueing system with a single removable and unreliable server whose arrivals form a Poisson process. Service times, repair times, and startup times are assumed to be generally distributed. When the queue length reaches N(N⩾1), the server is immediately turned on but is temporarily unavailable to serve the waiting customers. The server needs a startup time before providing service until there are no customers in the system. We analyze various system performance measures and investigate some designated known expected cost function per unit time to determine the optimal threshold N at a minimum cost. Sensitivity analysis is also studied.
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We consider the single-server constant retrial queue with a Poisson arrival process and exponential service and retrial times. This system has not waiting space, so the customers that find the server busy are forced to abandon the system, but they can leave their contact details. Hence, after a service completion, the server seeks for a customer among those that have unsuccessfully applied for service but left their contact details, at a constant retrial rate. We assume that the arriving customers that find the server busy decide whether to leave their contact details or to balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine the customers' behavior, and we identify the Nash equilibrium joining strategies. We also study the corresponding social and profit maximization problems. We consider separately the observable case where the customers get informed about the number of customers waiting for service and the unobservable case where they do not receive this information. Several extensions of the model are also discussed. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011