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Digital Object Identiﬁer 10.1109/ACCESS.2017.DOI

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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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

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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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

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 f−1

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

α−1

αf−1

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) = ρ2−ρ1qe

P U ρ2+ρ1qe

P U ρ2(µ/β)

λ(1 −ρ1qe

P U )(1 −ρ1qe

P U −ρ2q).(4)

Remark 3.1: If β≥α+ 1/f−1

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 =bβf−1

1(RP U )−1c,(5)

and

ne

SU =max njµ(f−1

2(RSU )−

ne

P U+1

P

i=1

PP U (i)¯

Bi)

1+α¯

B1k−1,0

o,(6)

in which

¯

Bj=j

β(1−ρ1)−ρne

P U−j+2

1(1−ρj

1)

β(1−ρ1)2, j = 1,2,· · ·,ne

P U+1,(7)

and

PP U (i) = (1 −ρ1)ρi

1

1−ρne

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 N≤ne

SU (k)and otherwise balk” is an

equilibrium, where ne

P U can be obtained from (5) and

ne

SU (k) = max µ(f−1

2(RSU )−¯

Bk)

1 + α¯

B1−1,0,(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

µ−λ.(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/(µ−λ), α + 1/f−1

1(RP U )) if µ−λ−

λαf−1

1(RP U )>0, and decreases with βin the interval

[max{α+ 1/f−1

1(RP U ), µα/(µ−λ)},+∞).

Proof: The equilibrium joining probability of PU requests,

qe

P U , is given in (2). Substituting (2) into (4), we get

WSU =

f−1

1(RP U)ρ2(β−µ+f−1

1(RP U)βµ)

λ(1−qβ f −1

1(RP U)ρ2

),β

α−1

αf−1

1(RP U)<1,

(β2−αβ+αµ)ρ2

λ(α−β)(α−β+qβρ2),β

α−1

αf−1

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(1−f−1

1(RP U )qβρ2)2,β

α−1

αf−1

1(RP U )<1,

αρ2Θ2

λ(α−β)2(α−β+qβρ2)2,β

α−1

αf−1

1(RP U )≥1,(12)

where Θ1=f−1

1(RP U )ρ2(1 + f−1

1(RP U )µ(1 −qρ2)) and

Θ2= (β−α)(α−β−2µ) + q(2β−α)µρ2. Obviously,

α, β, λ, ρ2, q , f−1

1(RP U )are positive. As stated in Section

IV, ρ1+ρ2<1. Then we easily get

1−qρ2>0,(13)

λβ +µα < µβ. (14)

If β/α −1/(αf −1

1(RP U )) <1, i.e., β < α + 1/f−1

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/(µ−λ), α + 1/f −1

1(RP U )) if µα/(µ−λ)<

α+ 1/f−1

1(RP U ), i.e., µ−λ−λαf −1

1(RP U )>0.

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Now we judge the monotonicity of WSU as β/α −

1/(αf−1

1(RP U )) ≥1. From (14), we get

Θ2=−(β−α)2−2µβ + 2µα + 2qβλ −qαλ

<−(β−α)2−2(λβ +µα)+2µα + 2qβλ −qαλ,

=−(β−α)2−2λβ(1 −q)−qαλ < 0.(15)

From (15), dWSU /dβ < 0if β/α −1/(αf −1

1(RP U )) ≥1,

i.e., β≥α+ 1/f−1

1(RP U ). Therefore, the mean sojourn

time of an arriving SU decreases with βin the interval

[max{α+ 1/f−1

1(RP U ), µα/(µ−λ)},+∞).

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 µ−λ−λαf−1

1(RP U )>0.

Remark 4.2 can be easily obtained from Proposition

4.1, since µ−λ−λαf−1

1(RP U )>0is identical to

α+ 1/f−1

1(RP U )> µα/(µ−λ). 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 µ−λ−λαf−1

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/f−1

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 β≥α+ 1/f−1

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

µ−λ−λαf−1

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

µ−λ−λαf−1

1(RP U )<0.(16)

As stated earlier in this subsection, we assume that the

steady condition is satisﬁed, so β > µα/(µ−λ). If (16)

holds, we have

α+1

f−1

1(RP U )<µα

µ−λ< 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 µ−λ−λαf−1

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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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 β→b−and

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,µ = 1, f−1

1(RP U )=3.

Tp(i, b+) if β→b+. PU adopts the threshold strategy ˜ne

P U

as β→b−and ˜ne

P U + 1 as β→b+. When βvaries from

b−to 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, f−1

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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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,µ = 1, f−1

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 Ωi={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

β∈Ωi

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

β−αqe

P U

.(24)

Lemma 5.2: In the no queue length information case, the

feasible interval can be written as

Ω0=

∅, f−1

1(RP U )< νlow ,

0, α+1

νlow , νlow ≤f−1

1(RP U )≤νup,

α+1

νup , α+1

νlow , f−1

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

f−1

1(RP U ),

β

α−1

αf−1

1(RP U ), β < α +1

f−1

1(RP U ).(26)

By (24) and (26), the mean sojourn time of joining PU

request is

We

P U,0(β) =

1

β−α, β ≥α+1

f−1

1(RP U ),

f−1

1(RP U ), β < α +1

f−1

1(RP U ).(27)

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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,

f−1

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

Ω0={β:νlow ≤We

P U,0(β)≤νup}. From (27), Ω0can

be rewritten as

Ω0={β|β < α+1

f−1

1(RP U )and νlow ≤f−1

1(RP U )≤νup}

∪{β|β≥α+1

f−1

1(RP U )and νlow ≤1

β−α≤νup}.(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

β∈Ω0

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

Ω0= [1.4,1.5] as α= 0.4, λ = 0.2, µ = 1, f−1

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, f−1

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(1−ρ1)(1−ρ1+ne

P U

1),(31)

where i= 1,2. The analytical solutions of both Ω1and

Ω2are 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

β∈Ω1

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 Ω1= [0.84,1.1], from Fig. 6 we ﬁnd in

the feasible interval Ω1there 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 Ω1= [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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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 Ω2, 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 ∀β∈Ω2,

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 Ω2= [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 β∈Ω2. Observing Fig. 7,

we ﬁnd

ne

SU (ˆ

β, j )≥ne

SU (β , j),for ∀β∈Ω2,

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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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