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Performance Modeling of Secondary Users in

CRNs with Heterogeneous Channels

Sharhabeel H. Alnabelsi Ahmed E. Kamal

Dept. of Electrical and Computer Eng., Iowa State University, Ames, Iowa 50011, USA

E-mail:{alnabsh,kamal}@iastate.edu

Abstract—The goal of this paper is to model heterogeneous

channel Access in Cognitive Radio Networks (CRNs). In CRNs,

when licensed users, known as Primary Users (PUs), are idle,

unlicensed users, known as Secondary Users (SUs) can use their

assigned channels. In the model we consider in this paper,

there are two types of licensed channels, where one type has

a larger bandwidth, and hence a higher service rate for SUs.

Therefore, SUs prefer to use such channels, if available, over

channels in the second type which have a lower service rate. SUs

may also switch from the second to the ﬁrst type of channels

when they become available, even if their current channels are

still available. Moreover in our performance model, we model

the SUs’ sensing process, and its dependence on the system

load, and number of sensing users. We use a Continuous Time

Markov Chain (CT M C ) modeling approach, and derive SUs’

performance metrics, which include SUs admission and blocking

probabilities, and their average waiting time in the system. We

also develop a baseline model in which SUs do not switch channels

between the two types, unless they are interrupted by PUs, and

compare its performance to our proposed model. Our numerical

analysis shows that our proposed model outperforms the baseline

model. We also, found that if sensing time is very small (≤1ms),

its effect on SUs performance is insigniﬁcant.

I. INT ROD UC TI ON

A. Background

Due to the temporal and spatial underutilization of licensed

spectrum bands, as well as the crowdedness of unlicensed

bands, a new spectrum access paradigm has been recently

proposed namely, Cognitive Radio (CR) [1]. CR enables users

to adjust their transceivers’ frequencies depending on the

availability of licensed frequency bands which are otherwise

unused by their licensees [2]. Thus, unlicensed wireless users,

called Secondary Users (SUs) can dynamically and oppor-

tunistically access unused licensed bands in order to improve

their throughput and service reliability. In this case, whenever

the licensed or the Primary Users (PUs) become active, SUs

must vacate their bands.

CRNs have many challenges such as spectrum sensing,

management, mobility, allocation and sharing [3], [4]. Usually,

SUs have QoS performance requirements, e.g., throughput and

maximum transmission delay. Evaluating these metrics is not

a trivial task, due to the CRNs dynamic nature, e.g., due to

PUs ﬂuctuating activities which may interrupt SUs, and hence

may need to access the channel multiple times just to ﬁnish

one communication session. To evaluate these performance

metrics, a few models have been proposed in literature. In

[5], [6], a Markovian model is proposed to analyze spectrum

access with and without buffering for new and interrupted

SUs requests, which is used to evaluate SUs mean waiting

time, and the probabilities of blocking, interruption, forced

termination, and non-completion. Results show that buffer-

ing SUs requests reduces SUs’ blocking and non-completion

probabilities, with a very small increase of forced termination

probability. However, in all other models network channels

are assumed symmetric in terms of channels bandwidth. In

addition, SUs’ sensing overhead is not considered in those

models.

A quasi-birth and death Markov chain with continuous time

and state space model is proposed in [7], to improve SUs

performance by distributing their ﬂows to multiple wireless

networks. Due to the high complexity for this model, an

approximation solution was proposed. They proposed two

admission control schemes for SU ﬂows priority, and no

priority schemes. In both admission control schemes, if an

SU is admitted to a network, it will not leave it until ﬁnishing

its transmission as long as it is not interrupted by a PU arrival.

A Markov model for spectrum sharing between PUs and

SUs is proposed in [8], when SUs are interrupted, they are

suspended and wait to access another channel in a call level

queue. During SUs’ suspension, packets generated by SUs

are either delayed or discarded, therefore the queue becomes

two sub-queues, delay and discard queues. Three metrics are

evaluated: packet loss ratio, packet delay, and throughput.

Results show by increasing SU suspension queue length, both

packet loss ratio and throughput increase and packet delay

decreases. A queuing network model for spectrum sharing

between PUs and SUs is introduced in [9], where a closed

form solution for equilibrium system state was derived as a

generating function. The model studies PUs QoS degradation

due to unreliable SUs spectrum sensing where an SU keeps

on using the channel, although a PU has arrived to the same

channel. Besides, an SU moves from its channel to another

only when it is interrupted by a PU arrival. Channels are

assumed symmetric in terms of service rate for PUs and SUs

in [8], [9].

B. Motivation

This work is motivated by:

•First, the fact that heterogeneous channels may be present

in the same locality, e.g., TV channels, cellular telephone

channels, wireless microphone channels, etc, which might

be used by SUs, if available. The bandwidth availability

in these channels is different. For instance, the licensed

spectrum of Wireless Microphones has 200 KHz band-

width, and that of Digital TV (ATSC) has 6 MHz band-

width. Therefore, this work is motivated by the possibility

of SUs switching channels opportunistically in order to

improve their performance.

•Second, the fact that some of the PUs of those channels

characterized by long idle times, e.g., Digital TV chan-

nels, which may lead to a sustainable SU throughput,

which also reduces channels switching overhead.

•Third, this work is also motivated by the need to consider

sensing time and its effect on channel utilization and

transmission delay. There are different technologies for

spectrum sensing such as energy and feature detection.

Energy detection sensing is frequent, and its typical

sensing time is less than 1 ms, while feature detection,

such as cyclostationary detection, is less frequent and

sensing time is around 24.2 ms for Digital TV [13].

We are therefore motivated to develop a modeling approach

that considers these three important issues and allows one

to evaluate the performance of SUs under these realistic

conditions.

C. Paper Contributions

The contributions of this paper are as follows:

1) We introduce a performance model for CRNs that models

heterogeneous channels, as well as the sensing process in

a manner that is dependent on the load. That is, the sens-

ing time increases if fewer channels become available,

and if fewer SUs are available to sense channels.

2) We introduce a strategy that gives preference to access

channels with potentially larger idle times and higher

bandwidth.

3) Through numerical results, we show that our proposed

strategy outperforms a baseline model that does not allow

switching between channels. In particular, our proposed

strategy reduces the mean waiting time for SUs in the

system.

4) Also, our numerical results show if sensing time is

very small (≤1ms), its effect on SUs performance is

insigniﬁcant.

D. Paper Organization

The rest of this paper is organized as follows. The model

and assumptions are explained in Section II. In Section III, our

proposed Continuous Time Markov Chains (CT M C) model

is presented. Performance metrics are derived in Section IV.

The baseline model is described in Section V, which is used

for comparison to our proposed model. Numerical results and

discussions are presented in Section VI. We conclude the paper

in Section VII.

II. MO DE L AN D ASS UM PT IO NS

We use a mixed queuing network to model the CRN system

where PUs are modeled as a closed chain of customers, while

SUs are modeled as an open chain. Table I shows the notation

and their description. Our proposed model contains two types

of channels, C1and C2, and a Virtual Queue (VQ), which

is used to accommodate SUs. It is assumed that channels in

C1have a higher bandwidth than channels in C2, and SUs,

therefore, prefer to use channels in C1. SUs may move from

the VQ to a channel in C1, if available, as their ﬁrst preference.

TABLE I

TABL E OF NOTATIO NS ,WHERE i={1,2}.

Notation Description

CiNumber of channels in type i.

vNumber of SUs in the system: VQ, C1, and

C2.

βThe maximum size of the VQ buffer.

sA binary variable for sensing state, where

0 means no sensing is being conducted. 1,

otherwise.

piNumber of busy PUs in type ichannels.

ηSUs sensing rate.

Ψ(p1+p2,v)SUs sensing rate function.

pfProbability of false alarm.

λsSUs arrival rate.

λpiPUs arrival rate in type ichannels.

µsiSUs service rate in type ichannels.

µpiPUs service rate in type ichannels.

{v, p1, p2, s}A system state where v,p1,p2, and sare the

number of SUs in system, busy PUs in C1,

busy PUs in C2, and sensing state.

πv,p1,p2,s The probability of a steady state {v, p1, p2, s}.

pbProbability of SUs blocking, for C1and C2

overall.

paProbability of SUs admission, for C1and C2

overall.

LAverage number of SUs in the system: VQ,

C1, and C2.

WSUs average waiting time in the system: VQ,

C1, and C2, until ﬁnishing their packet trans-

mission.

Otherwise, they move to a channel in C2, if available. In

this model, whenever an SU detects an available channel in

C1, the SU starts using this channel, although the SU may

have been using a channel in C2. The purpose for doing so

is to improve the SUs performance. SUs detect out-of-band

channels availability by exchanging control information over

Common Control Channel (CCC) [10]. The maximum number

of SUs which can be in the system equals the VQ buffer size,

β. Therefore, if an SU is interrupted during its service by the

channel’s PU, the SU moves to the VQ, and starts to sense

for available channels in order to ﬁnish its own transmission,

and then leaves the system.

A. Assumptions

◦SUs exchange channels’ information, such as channel’s

availability using a CCC [11], which is assumed to be

always available.

◦Each channel has its own PU assigned to it.

◦The number of SUs in the network is unlimited, but a

maximum of βcan be in the system, either occupying

channels, or waiting for channels to become available.

◦Each channel is modeled as a server with no buffer.

◦There are two types of channels, type 1and type 2.

◦SUs assign a higher priority for using type 1channels over

type 2, due to the higher throughput of type 1channels.

◦Assume each SU has two transceivers, one for data trans-

mission, and the other for exchanging control packets with

other SUs over the CCC, and for conducting in band and

out-of-band sensing.

◦The Virtual Queue (VQ) is a concept to hold the newly

arrived and interrupted SUs, as well as SUs being served.

◦When an SU ﬁnishes its transmission, the SU leaves the

network. However, if an SU is interrupted, the SU moves

to the VQ and waits for an available channel in order to

complete its transmission.

◦When a PU or an SU ﬁnishes its transmission on a type 1

channel, e.g., channel k, then an SU being served on a type

2 channel, if any, moves to channel k, in order to improve

the SUs throughput.

◦SUs sensing time1is exponentially distributed with a rate

that is dependent on v,p1, and p2. Let Ψ(p1+p2,v)be this

sensing rate function, and it will be deﬁned in the numer-

ical results Section, Section VI, equation (13). Practically,

Ψ(p1+p2,v)>> λs,λp1,λp2,µs1,µs2,µp1, and µp2.

◦Sensing is triggered when an SU arrives, given there is an

idle channel. Or, when a PU or an SU ﬁnishes transmission

and there is at least one SU waiting in the VQ.

◦We only model good sensing which results in ﬁnding an idle

channel that the SU can use. Modeling sensing which does

not result in accessing a channel, either because all channels

are busy, or because there are no waiting SUs, will have no

bearing on the system operation, and does not change the

model.

◦The probability of misdetection under sensing is assumed to

be very small, and is therefore negligible, in order to reduce

the model complexity. Misdetection is deﬁned as detecting

the channel as idle, while the channel is occupied by a PU’s

transmission.

◦The probability of false alarm, pf, is considered in this

model, since it has much higher effect than the probability

of misdetection in our system model, and it is usually less

than 0.1, e.g., as in IEEE 802.22 CRNs standard [12]. False

alarm is deﬁned as detecting the channel as busy by a PU’s

transmission, while in reality it is idle.

B. Parameters

◦C1and C2are the number of channels of types 1and 2,

respectively, and are also the number of PUs assigned to

these channels.

◦PUs assigned to type 1and type 2channels, have exponen-

tially distributed inter-arrival times with rates of λp1and

λp2, respectively, when they are idle.

◦PUs using type 1and type 2channels have service rates

of µp1and µp2, respectively, with exponential distributions,

when they are active.

◦βis the maximum size of the VQ2.

1In this paper, it is assumed that what we refer to as the sensing time,

includes both the channel sensing, and channel switching times.

2In our numerical results, βis set to a large value such that SUs pa≈1.

◦SUs arrival rates to the VQ is λswith Poisson distributions.

◦On type 1and type 2channels, the SUs service rates are

µs1and µs2, respectively, with an exponential distributions.

C. Variables

◦sis a binary variable, {0,1}, for sensing state, where 0

means no sensing is being conducted. While 1, otherwise.

◦vis the total number of SUs in the VQ, including those

being served by types 1and 2channels, while in this model

vis taken as ﬁnite, the virtual queue size, β, can also be

set to a very large number, hence approximating the inﬁnite

number of SUs case, as will be shown in Section VI.

◦p1and p2are the numbers of PUs being served by type 1

and type 2channels, respectively.

◦For our model, we deﬁne the state space, call it ζ, as

(v, p1, p2, s), where: 0≤v≤β,0≤p1≤C1,

0≤p2≤C2, and sis a binary variable, such that vmust

be ≥1, if s= 1, which means at least one SU must exist

to conduct sensing.

◦let πˆv, ˆp1,ˆp2,s be the stationary probability vector where v=

ˆv,p1= ˆp1,p2= ˆp2, and sis a binary variable such that if

s= 1, sensing is being conducted by SU(s). Otherwise; no

sensing.

It is to be noted that the assumption of memoryless

distributions, i.e., exponential distributions, has been made

in order to make the model mathematically tractable. This

is a standard assumption that is made in such complicated

models. It is to be also noted that the queuing network

is ergodic, because it is irreducible and has a ﬁnite state

space. The queueing network therefore has a unique steady

state (S.S.) distribution, −→

π.

III. MOD EL FO RM UL ATIO N

There are 5 cases of the global balance equations. Since

sensing is inconsequential when no channels are available, it

is assumed that sensing is terminated when a PU arrives to

occupy its channel and no other channels are available. It is

also assumed that at most one decision can be made based on

sensing at the same time.

In order to model the system exactly, a greater number of

state variables need to be included, which will signiﬁcantly

increase the system complexity. Therefore, we introduce two

relaxations which result on bounds on system performance.

These are an optimistic bound and a pessimistic bound. The

deﬁnition of these two models are as follows:

•For the optimistic bound analysis, if sensing is con-

ducted, then we assume it is on a channel in type 2(lower

SUs service rate), i.e., all available channels in type 1are

being used.

•For the pessimistic bound analysis, if sensing is con-

ducted, then we assume it is on a channel in type 1(higher

SUs service rate), i.e., all available channels in type 2are

being used.

Throughout this section, we consider the optimistic bound

performance, while formulating the global balance equations.

With minor modiﬁcations of these global balance equations,

we can also model the pessimistic bound performance.

Case 1: If v < (C1−p1), then all active SUs are using

channels in C1. Hence, the global balance equations are

equations (1) and (2).

πv,p1,p2,1[λs+vµs1+p1µp1+p2µp2+ (C1−p1)λp1+

(C2−p2)λp2+ Ψ(p1+p2,v)(1 −pf)]

=πv−1,p1,p2,0[λs]v≥1+πv−1,p1,p2,1

[λs]v≥2+πv+1,p1,p2,1[(v+ 1)µs1] + πv,p1+1,p2,1

[(p1+ 1)µp1]v≥1+πv,p1,p2+1,1[(p2+ 1)µp2]v≥1

+πv,p1,p2−1,1[(C2−p2+ 1)λp2]v≥1

+v

C1−p1+ 1πv,p1−1,p2,0[(C1−p1+ 1)λp1]

+C1−p1+ 1 −v

C1−p1+ 1 πv,p1−1,p2,1[(C1−p1+ 1)λp1]v≥1.

(1)

Sensing is considered for all channels, with channels in C1

given a higher priority when sensed by SUs. In equation (1),

v≥xis an indicator function which equals 1if the condition,

v≥x, holds. Otherwise, it is 0. The LHS of equation (1),

is the probability ﬂux of leaving state (v, p1, p2,1) due to: an

SU arrival with rate λs, an SU in C1ﬁnishes its transmission

with rate vµs1, a PU in C1or C2ﬁnishing its transmission

with rates p1µp1and p2µp2, respectively, a PU arrived to C1

or C2with rates (C1−p1)λp1or (C2−p2)λp2, respectively,

and end of sensing with rate of Ψ(p1+p2,v)(1 −pf).

The RHS of equation 1, is probability ﬂux of entering state

(v, p1, p2,1). This is due to: an SU arrived while the system in

states (v−1, p1, p2,0) and (v−1, p1, p2,1). An SU ﬁnishing its

transmission while the system in state (v+ 1, p1, p2,1) with

rate (v+ 1)µs1, a PU completing service while the system

is in state (v, p1+ 1, p2,1) or (v, p1, p2+ 1,1), with rates

(p1+ 1)µp1or (p2+ 1)µp2, respectively, a PU arriving to

C2with rate (C2−p2+ 1)λp2, while the system in state

in state (v, p1, p2−1,1), a PU arriving to C1while in state

(v, p1−1, p2,0), and interrupting an SU which is using its

channel, thus sensing by the SU is triggered with probability

v

C1−p1+1 , and a PU arriving to its channel which is not being

used by an SU, with probability C1−p1+1−v

C1−p1+1 , while the system

is in state (v, p1−1, p2,1), given there has been sensing. In

the rest of the paper, only the new transition states will be

explained, due to space limitation.

In equation (2), the LHS is similar to that in equation (1),

but there is no sensing. In the RHS, the second term to the

last, the system transits from state (v, p1−1, p2,0) to state

(v, p1, p2,0), due to a PU arrival to its channel in C1where

no SU exits, with a probability of C1−p1+1−v

C1−p1+1 , given there was

no sensing.

Case 2: If (C1−p1)≤v < (C1−p1) + (C2−p2),

then the global balance equations are (3) and (4). In this case,

if PU being served within C1ﬁnishes its transmission, SUs

sensing is triggered. Also, these equations implicitly model

the SUs preference to be served by C1channels rather than

C2channels. Thus, when a PU in C1ﬁnishes its transmission,

say at channel k, sensing is triggered, and then an SU moves

to channel k.

πv,p1,p2,0[λs+vµs1+p1µp1+p2µp2+ (C1−p1)λp1+ (C2

−p2)λp2] = πv,p1,p2,1Ψ(p1+p2,v)(1 −pf)v≥1

+πv+1,p1,p2,0[(v+ 1)µs1] + πv,p1+1,p2,0[(p1+ 1)µp1]

+πv,p1,p2+1,0[(p2+ 1)µp2]

+C1−p1+ 1 −v

C1−p1+ 1 πv,p1−1,p2,0[(C1−p1+ 1)λp1]

+πv,p1,p2−1,0[(C2−p2+ 1)λp2].

(2)

In equation (3) in the last term on the RHS, a PU arrives

to its channel in C2, where no SU is using it, with probability

C2−p2+1−(v−C1−p1)

C2−p2+1 , given sensing was not being conducted.

Thus, system transits to state (v, p1, p2,0) (LHS). In equation

(4), the system transits to state (v, p1, p2,1) in LHS, from

different states, for example: from state (v, p1, p2−1,0) with

a probability of v−C1−p1

C2−p2+1 , when a PU arrives to C2and

interrupts an SU that is using its channel. Thus, the PU arrival

causes sensing to start. The same thing occurs in the second

term to last, in state (v, p1, p2−1,1) with the same probability.

However, in this case an SU which was already engaged in

sensing will just continue to sense. However, in the last term

a PU arrives to its channel in C2, where no SU is using it,

with probability C2−p2+1−(v−C1−p1)

C2−p2+1 , given sensing was being

conducted.

Case 3: If v= (C1−p1) + (C2−p2), then the global

balance equations are equations (5) and (6). In equation (5)

the last term in RHS, shows sensing is triggered by a PU

interruption of an SU which was served by the PU’s channel,

with probability C2−p2

C2−p2+1 , given there is still one free channel

in C2. However, in equation (6), the last term on the RHS

corresponds to a PU arriving to a channel where no SU was

being served, with probability of 1

C2−p2+1 , and hence, sensing

is not triggered.

Case 4: If (C1−p1) + (C2−p2)< v < β, then, equations

(7) and (8) are the global balance equations. Recall that for

the optimistic bound analysis, if sensing is conducted, then it

is at a channel in C2, i.e., all available channels in C1are

being used.

πv,p1,p2,0[λs+ (C1−p1)µs1+ (v−C2+p2)µs2+p1µp1

+p2µp2+ (C1−p1)λp1+ (C2−p2)λp2]

=πv,p1,p2,1Ψ(p1+p2,v)(1 −pf)v≥1

+πv+1,p1,p2,0[(v+ 1 −C1+p1)µs2] + πv,p1,p2+1,0

[(p2+ 1)µp2] + C2−p2+ 1 −(v−C1+p1)

C2−p2+ 1

πv,p1,p2−1,0[(C2−p2+ 1)λp2].

(3)

πv,p1,p2,1[λs+ (C1−p1)µs1+ (v−C1+p1−1

(v−C1+p1)≥1)µs2+p1µp1+p2µp2+ (C1−p1)λp1+ (C2

−p2)λp2+ Ψ(p1+p2,v)(1 −pf)] = πv−1,p1,p2,0[λs]v≥1

+πv−1,p1,p2,1[λs]v≥2+πv+1,p1,p2,0[(C1−p1)µs1]

+πv+1,p1,p2,1[(C1−p1)µs1] + πv+1,p1,p2,1

[(v+ 1 −C1+p1−1)µs2] + πv,p1+1,p2,0[(p1+ 1)µp1]

+πv,p1+1,p2,1[(p1+ 1)µp1]v≥1+πv,p1,p2+1,1

[(p2+ 1)µp2]v≥1+πv,p1−1,p2,1[(C1−p1+ 1)λp1]v≥1

+πv,p1−1,p2,0[(C1−p1+ 1)λp1] + v−C1+p1

C2−p2+ 1

πv,p1,p2−1,0[(C2−p2+ 1)λp2] + v−C1+p1

C2−p2+ 1 πv,p1,p2−1,1

[(C2−p2+ 1)λp2]v≥1+C2−p2+ 1 −(v−C1+p1)

C2−p2+ 1

πv,p1,p2−1,1[(C2−p2+ 1)λp2]v≥1.

(4)

πv,p1,p2,1[λs+ (C1−p1)µs1+ (C2−p2−1)µs2C2−p2̸=0

+p1µp1+p2µp2+ (C1−p1)λp1+ (C2−p2)λp2+

Ψ(p1+p2,v)(1 −pf)]

=πv−1,p1,p2,0[λs]v≥1+πv−1,p1,p2,1[λs]v≥2

+πv+1,p1,p2,0[(C1−p1)µs1] + πv+1,p1,p2,0[(C2−p2)µs2]

+πv+1,p1,p2,1[(C1−p1)µs1]

+πv+1,p1,p2,1(C2−p2−1(C2−p2)≥1)µs2

+πv,p1+1,p2,0[(p1+ 1)µp1] + πv,p1,p2+1,0[(p2+ 1)µp2]

+πv,p1+1,p2,1[(p1+ 1)µp1]v≥1+πv,p1,p2+1,1

[(p2+ 1)µp2]v≥1+πv,p1−1,p2,0[(C1−p1+ 1)λp1]

+πv,p1−1,p2,1[(C1−p1+ 1)λp1]v≥1

+πv,p1,p2−1,1[(C2−p2+ 1)λp2]v≥1

+C2−p2

C2−p2+ 1πv,p1,p2−1,0[(C2−p2+ 1)λp2].

(5)

πv,p1,p2,0[λs+ (C1−p1)µs1+ (C2−p2)µs2+p1µp1

+p2µp2+ (C1−p1)λp1+ (C2−p2)λp2]

=πv,p1,p2,1Ψ(p1+p2,v)(1 −pf)v≥1

+1

C2−p2+ 1πv,p1,p2−1,0[(C2−p2+ 1)λp2].

(6)

In equations (7) and (9), the last term in RHS corresponds

to a PU arriving to a channel where an SU is sensing it,

which occurs with probability of 1

C2−p2+1 , hence sensing is

terminated. However, in Equation (8) the last term in RHS,

shows that the sensing has not been terminated , since the PU

arrives to a channel where sensing is not being conducted,

with probability of C2−p2

C2−p2+1 . However, the PU arrival causes

an SU interruption, where the SU goes back to the VQ, and

waits for a channel to become available. Recall that we assume

sensing is always conducted at a channel in type 2.

Case 5: In this case v=β. As a result, equations (9) and

(10) are the global balance equations.

πv,p1,p2,0[λs+ (C1−p1)µs1+ (C2−p2)µs2+p1µp1

+p2µp2+ (C1−p1)λp1+ (C2−p2)λp2]

=πv,p1,p2,1Ψ(p1+p2,v)(1 −pf)v≥1

+πv−1,p1,p2,0[λs] + πv,p1−1,p2,0[(C1−p1+ 1)λp1]

+πv,p1,p2−1,0[(C2−p2+ 1)λp2]

+1

C2−p2+ 1πv,p1,p2−1,1[(C2−p2+ 1)λp2]v≥1.

(7)

πv,p1,p2,1[λs+ (C1−p1)µs1+ (C2−p2−1

(C2−p2)̸=0)µs2+p1µp1+p2µp2+ (C1−p1)λp1

+ (C2−p2)λp2+ Ψ(p1+p2,v)(1 −pf)]

=πv−1,p1,p2,1[λs]v≥2+πv+1,p1,p2,1[(C1−p1)µs1]

+πv+1,p1,p2,1(C2−p2−1(C2−p2)≥1))µs2

+πv+1,p1,p2,0[(C1−p1)µs1] + πv+1,p1,p2,0

[(C2−p2)µs2] + πv,p1+1,p2,1[(p1+ 1)µp1]v≥1

+πv,p1,p2+1,1[(p2+ 1)µp2]v≥1

+πv,p1+1,p2,0[(p1+ 1)µp1] + πv,p1,p2+1,0

[(p2+ 1)µp2] + πv,p1−1,p2,1[(C1−p1+ 1)λp1]v≥1

+C2−p2

C2−p2+ 1πv,p1,p2−1,1[(C2−p2+ 1)λp2]v≥1.

(8)

πβ,p1,p2,0[(C1−p1)µs1+ (C2−p2)µs2+p1µp1+

p2µp2+ (C1−p1)λp1+ (C2−p2)λp2]

=πβ−1,p1,p2,0[λs]β≥1+πβ,p1,p2,1

Ψ(p1+p2,v)(1 −pf)β≥1+πβ,p1−1,p2,0

[(C1−p1+ 1)λp1] + πβ,p1,p2−1,0[(C2−p2+ 1)λp2]

+1

C2−p2+ 1πβ,p1,p2−1,1[(C2−p2+ 1)λp2]β≥1.

(9)

πβ,p1,p2,1[(C1−p1)µs1+ (C2−p2−1(C2−p2)̸=0)µs2

+p1µp1+p2µp2+ (C1−p1)λp1+ (C2−p2)λp2+

Ψ(p1+p2,v)(1 −pf)]

=πβ−1,p1,p2,1[λs]β≥2+πβ,p1+1,p2,1[(p1+ 1)µp1]

β≥1+πβ,p1,p2+1,1[(p2+ 1)µp2]β≥1

+πβ,p1+1,p2,0[(p1+ 1)µp1] + πβ,p1,p2+1,0

[(p2+ 1)µp2] + πβ,p1−1,p2,1[(C1−p1+ 1)λp1]β≥1

+C2−p2

C2−p2+ 1πβ,p1,p2−1,1[(C2−p2+ 1)λp2]β≥1.

(10)

IV. PER FO RM AN CE ME TR IC S

In this section, we introduce several performance metrics

which can be used to evaluate CRN performance. These

include the probabilities of admission and blocking of SUs,

average number of SUs in the system during the network

operation, and average waiting time for SUs in the system

until completing service. We solved the steady state probability

distribution, −→

π, by solving the equation −→

π Q = 0, where Q

is the transition rate matrix that can be constructed using the

global balance equations (1)−(10).

However, the number of linearly independent global balance

equation is (m−1). Therefore, use the fact that the summation

of all probabilities in the steady state distribution equals 1. As

a results, we have mlinearly independent solvable equations.

Let us introduce the following deﬁnitions:

Deﬁnition IV.1 Probability of blocking of SUs (pb): It is the

probability that a new SU request for transmission is blocked

due to the lack of space in the VQ.

Deﬁnition IV.2 Probability of admission for SUs (pa): It is

the probability that a new SU request for transmission is

admitted.

Deﬁnition IV.3 The average number of SUs in the system (L),

which includes those being served by channels of types 1and

2, and also those waiting for a channel to become available.

Deﬁnition IV.4 Average waiting time (W) of SUs, which

is measured from the instant of arrival, until ﬁnishing its

transmission.

The following equations are used to evaluate the perfor-

mance metrics of our proposed model.

1) The probability of blocking for SUs (pb) is given by

equation (11).

pb=

C1

p1=0

C2

p2=0

1

s=0

{v=β},

(v,p1,p2,s)∈ζ

πv,p1,p2,s.(11)

2) The probability of SU admission, (pa), equals to 1−pb.

3) The average number of SUs in the system (L), is given

by equation (12).

L=

C1

p1=0

C2

p2=0

1

s=0β

v=1,

(v,p1,p2,s)∈ζ

v×πv,p1,p2,s.(12)

4) To ﬁnd the average waiting time W, we appeal to Little’s

Theorem, where Lis given by equation (12), and Wis

expressed as W=L

pa×λs.

V. BASELINE MODEL

In this section, we introduce and model another system.

This is a system similar to our proposed model, but with no

channel switching to type 1 channels (if available) by SUs

which are being served in type 2 channels, unless there are no

longer available channels on type 2. We developed this system

and use it as a baseline model to establish the advantages of

our proposed approach. For example, if an SU arrives and

selects a channel, say from set C2, the SU keeps using this

channel, until ﬁnishing its transmission, as long as this channel

is available. However, if the SU is interrupted, and sense there

are no available channels on type 2 to use it, and there is

an available channel on type 1, therefore the SU switches to

this channel. Otherwise, when no channel is available in both

types 1 and 2, the SU is buffered in the VQ, until a channel

becomes available. Due to space limitation in this paper, we

did not write the global balance equations for the baseline

model. The performance metrics for the baseline model are

similar to our model, in Section IV. Similarly, those equations

with minor modiﬁcations are used to evaluate the baseline

model performance metrics, and due to space limitation in

this report, we did not rewrite these equations.

VI. NUMERICAL RE SU LTS

This section presents the numerical results for SUs average

waiting time, with respect to SUs arrival rate to the system, λs.

Also, we study the effect of SUs sensing rate on the system

performance.

The sensing rate is dependent on both the number of unused

channels, and on the number of SUs performing the sensing

process. It was proven in [14] that the expected time to detect

an unused channel is inversely proportional to the number

of unused channels, which means that the sensing rate is

proportional to this number. Moreover, if the total number

of channels very large, and is evenly divided among the SUs

sensing for available channels (out-of-band sensing), then the

rate of detecting an empty channel is the sum of the individual

SUs sensing rates. We therefore express the sensing rate, as

a function of p1,p2, and v,Ψ(p1+p2,v), as shown in equation

(13), where ηis the sensing rate when there is only a single

SU sensing, and there is only one available channel.

Ψ(p1+p2,v)=ηˆ

Nˆ

I. (13)

ˆ

Nis the number of SUs in the system which are not being

served by channels (waiting/interrupted) or want to improve

their performance by switching to a channel in type 1, and

therefore conduct out-of-band sensing. ˆ

Iis number of idle

channels in type 1and 2channels which are not being used by

SUs or PUs. Based on the global balance equations (1)−(10),

if v≤(C1−p1),ˆ

N= 0 (No SUs are interested in sensing,

since all current SUs are being served by type 1channels). If

(C1−p1)< v ≤(C1−p1+C2−p2), then ˆ

N=v−(C1−p1).

Otherwise, ˆ

N=v−(C1−p1+C2−p2). If v≤(C1−p1+

C2−p2), then ˆ

I= (C1−p1+C2−p2)−v. Otherwise,

ˆ

I= 1. According to equation (13) the sensing rate increases

(the sensing time decreases) when more SUs are active and

sensing the channels. The sensing rate decreases (the sensing

time increases) when more PUs are active, and therefore there

are fewer available channels, and it takes longer to search for

and sense those channels.

In order to evaluate our proposed model, we consider two

different scenarios as follows, it is worth mentioning that in

both scenarios, SUs service rates in type 1and 2channels are

different.

◦Scenario 1parameters, µs1= 60,µs2= 15,λp1=λp2=

5,µp1=µp2= 10,η= 250,C1=C2= 4,pf= 0.09,

and β= 40.

◦Scenario 2parameters, µs1= 80,µs2= 20,λp1= 5,

λp2= 25,µp1= 15,µp2= 80,η= 380,C1=C2= 3,

pf= 0.05, and β= 50. This scenario, is different from

the ﬁrst one, where PUs arrival and service rates are not

equal in both channel types. Also, PUs service time in type

1channels is greater than those in type 2, to capture the

heterogeneity nature, such as in TV channels and cellular

phones channels.

A. Average waiting time of SUs:

We show how the SUs’ average waiting time, W, changes

with respect to SUs arrival rate, λs.It is worth mentioning

the probability of SUs admission, pa, in our model and

the baseline model for all numerical results in this section

SUs average waiting time (sec)

SUs arrival rate ( s)

Our model, OB

Baseline model, OB

Our model, PB

Baseline model, PB

Fig. 1. Scenario 1, SUs Wtime with respect to

their arrival rate, λs.

SUs average waiting time (sec)

SUs arrival rate ( s)

Our model, OB

Baseline model, OB

Our Model, PB

Baseline model, PB

Fig. 2. Scenario 2, SUs Wtime with respect to

their arrival rate, λs.

SUs average waiting time (sec)

sensing rate ( )

Scenario 1, PB

Scenario 2, OB

Fig. 3. SUs Wtime, with respect to their sensing

rate, η, for scenarios 1and 2for PB and OB,

respectively, where λsis ﬁxed and set to 100.

is almost 1. In our results, βis set to a value, e.g., in

scenario 2β= 50, such that pais almost 1. We have varied

βsize up to 100 in both Scenarios studies, however, this

do not change the numerical results, e.g., W. Therefore,

we approximate the inﬁnite number of SUs case in our

results. Figure 1 which corresponds to scenario 1, shows that

Wincreases by increasing SUs arrival rate, λs. Our model

outperforms the baseline model in both the Optimistic Bound

(OB) and Pessimistic Bound (PB) analysis, because our model

reduces Wfor SUs in the system. For example, for the OB

analysis, and when λs= 20,100, and 150, our model reduces

Wby up to 12.44%,20.68%, and 11.99%, respectively, with

respect to the baseline model. One observation, when λs=

100,Wreduction is higher than when λs= 150. Therefore,

our model Wreduction percentage over the baseline model

reaches its maximum value, when λsincreased to some value,

and then this percentage decreases.

Also, Figure 2, which corresponds to scenario 2system

parameters, shows that Wincreases by increasing λs. This

ﬁgure, shows although the PUs behavior is different between

the two types of channels, our model outperforms the baseline

model in the OB and PB analysis. For example, for the PB

analysis, and when λs= 50, our model reduces Wby up to

16.23% with respect to the baseline model. Please notice that

sensing rate, equation (13), is higher than the SUs and PUs

arrival and service rates in these cases studies.

B. Sensing Rate:

We consider the Pessimistic and the Optimistic bounds

analysis for scenarios 1and 2, respectively, in our model to

study the effect of sensing rate, η, on SUs performance. The

system parameters correspond to scenarios 1and 2parameters,

except that in both scenarios λsis ﬁxed and is set to 100, while

the sensing rate is varied on the X axis. Figure 3 shows that

the SUs’ average waiting time, W, decreases by increasing η.

The smallest value for sensing rate in this ﬁgure is 50, i.e., an

average sensing time of 20 ms, which is about the sensing time

using feature detection [13]. However, for the energy detection

method, the sensing time is ≤1ms [13], or η≥1000. Clearly,

when the energy detection method is used instead of feature

detection, the SUs performance is better and Wdecreases.

This ﬁgure also shows that if ηis increased beyond 1000,

its effect on SUs performance is insigniﬁcant for both cases

studies scenarios, e.g., in scenario 1for the PB, when ηis

increased from 50 to 1000,Wis reduced by up to 34.56%,

however, when ηis increased from 1000 to 2000,Wis only

reduced by up to 3.42%. Since our model discards SU arrivals

occurring while channels are being sensed, Win Figure 3 is

underestimated.

VII. CONCLUSIONS

In this paper we proposed a model for heterogeneous

channel access in Cognitive Radio Networks (CRNs). In this

model, there are two types of licensed channels, where one

type has a larger bandwidth. SUs may use the ﬁrst type if

it is available, or if it becomes available. We also model the

SUs’ sensing process and study its effect on performance, such

that sensing rate is dependent on both the number of unused

channels, and on the number of SUs performing the sensing

process. We used a mixed queuing network model to model

the CRN system, and developed the global balance equations

for a CT M C . We derived SUs’ performance metrics, such as

SUs admission and blocking probabilities, and their average

waiting time in the system. We compare our proposed system

to a baseline model, which is the same as our proposed model,

except that SUs in type 2 channel can not improve their

throughput by switching to channels in type 1, if available,

unless they are interrupted at their current type 2 channels.

Numerical results show that our proposed model outperforms

the baseline model. We also found that if sensing time is very

small (≤1 ms), its effect on SUs performance is insigniﬁcant.

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