Wireless Sensor Network, 2009, 1, 397-406
doi:10.4236/wsn.2009.15048 Published Online December 2009 (http://www.scirp.org/journal/wsn).
397
Modeling and Analysis of Random Periodic Spectrum
Sensing for Cognitive Radio Networks
Caili GUO, Zhiming ZENG, Chunyan FENG, Qi LIU
School of Information and Communication Engineering,
Beijing University of Posts and Telecommunications, Beijing, China
Received June 18, 2009; revised August 20, 2009; accepted August 21, 2009
Abstract
A random periodic spectrum sensing scheme is proposed for cognitive radio networks. The sensing period,
the transmission time for primary users and cognitive radios are extended to general forms as random vari-
ables. A generalized Markov analytical model for sensing period optimization is presented, and the applica-
tions of the proposed analytical model by using examples involving primary user systems with both voice
and data traffic are illustrated. The analysis and numerical results show that sensing period does affect the
maximum rewards of the channel, and the analytical model is justified by its flexibility since it uses general
forms of the sensing period, the transmission time for primary users and cognitive radios. Hence the model
can be easily adapted for the analysis of many different applications.
Keywords: Cognitive Radio Networks, Random Periodic Spectrum Sensing, Generalized Markov Process,
the Optimal Sensing Period
1. Introduction
Due to energy consumption and hardware implication of
Cognitive Radios (CRs), it is undesirable and impractical
to assume the spectrum sensing to be continuous. In a
practical CR network, such as an IEEE 802.22 network
[1], a periodic spectrum sensing scheme where the spec-
trum is sensed periodically to determining the pres-
ence/absence of Primary Users (PUs) is preferable. The
sensing time and sensing period are two key sensing pa-
rameters for periodic sensing scheme. The former is a
pre-defined amount of time used to achieve the desirable
level of detection quality and is mainly depended on
PHY-layer sensing methods, such as energy detection,
matched filter and feature detection. And the latter de-
fined as the interval between two successive detection
processes has a significant impact on the sensing effi-
ciency of CRs. In the case of the sensing period is rela-
tively large, both some opportunities may go undiscov-
ered and interference to PUs may occur, whereas blindly
reducing the sensing period is not desirable either, as it
increases the sensing overhead. Thus the design of any
periodic sensing scheme involves balancing the tradeoffs
among spectrum utilization, interference to PUs, and
sensing overhead by selecting an appropriate sensing
period. We usually consider a spectrum consist of several
channels, and each channel can be a frequency band with
certain bandwidth, a spreading code in a CDMA network
or a set of tones in an OFDM system. Here we use the
term channel broadly.
In CR networks, the control of quiet period, during
which all CRs should suspend their transmissions so that
any CR monitoring the channel may observe the pres-
ence/absence of PU signals without interference, can be
synchronous or asynchronous [1–2]. Accordingly there
are two kinds of periodic sensing schemes: One is syn-
chronous sensing period and the other is asynchronous
sensing period. Most of the existing works focused on
the synchronous sensing period schemes [3–4]. As a
simple solution for design and implementation, the syn-
chronous sensing period scheme sets a pre-determined
fixed sensing period for all channels. While it does not
need the scheduling of quiet period for each channel
among CRs, it shows less flexible. Recent researches
[5–7] showed that the asynchronous sensing period
scheme is more favorable, in which sensing period can
be adjusted adaptively according to the channel-usage
characteristics of each channel by the MAC-layer sens-
ing protocol or through a dedicated control channel [8].
Kim [5] proposed an adaptation algorithm in which the
The material in this paper is based on “Random Periodic Spectrum
Sensing with Sensing Period Optimization for Cognitive Radio Net-
works”, by Caili Guo, Zhiming Zeng, Chunyan Feng and Qi Liu which
appeared in the proceedings of 11th IEEE International Conference on
Communications Systems, ICCS 2008, Guangzhou, China, November
2008.

C. L. GUO ET AL. 398
optimal sensing period is uniquely determined for each
channel to maximize the discovery of opportunities as
well as minimize the delay in locating an idle channel.
However, this approach clearly did not consider the im-
pact of sensing period selection on interference to PUs.
In [6], we extended [5] to a Flexible Sensing Period
(FSP) mechanism that introduces the “period control
factor” to control each channel’s sensing period adap-
tively to tradeoff undiscovered opportunities and inter-
ference to PUs with sensing overhead effectively, but as
far as each channel is concerned, the FSP also consider
that sensing period is still fixed. In order to combat the
fluctuation of sensing period induced by the varying of
channel-usage characteristics, in [7] we described a
Fuzzy Spectrum Sensing Period Optimization (FSSPO)
algorithm where each channel’s sensing period is adap-
tively adjusted in real time with fuzzy logic and parame-
ters optimization.
Existing approaches of asynchronous sensing period in
[5,6] and [7] only considered how to adjust sensing pe-
riod which is usually regarded as a constant once it is
determined, and they all assumed that the sensing results
are perfect. In this paper, the random periodic sensing
scheme we proposed extends the sensing period, the time
of transmission for primary users and cognitive radios to
random variables and more practical situation where
sensing error exists is considered. As the PUs have the
highest priority, we also introduce a back-off mechanism
where a random back-off time is generated whenever
PUs release the channel and CRs have to delay for the
back-off time before occupying the channel. Here we
focus on how to model the proposed random periodic
sensing scheme to a generalized Markov process and
how to derive the optimal sensing period. To support the
proposed analytical model for sensing period optimiza-
tion, we also illustrate the applications of the analytical
model by using examples involving PU systems with
both voice and data traffic.
The rest of the paper is organized as follows. Section 2
introduces the random periodic spectrum sensing scheme.
A generalized Markov analytical model for sensing pe-
riod optimization is constructed in Section 3. Then in
Section 4, how to obtain performance measures of chan-
nels is considered. Example applications of the proposed
analytical model for real networks are illustrated in Sec-
tion 5. Numerical examples are presented and discussed
in Section 6. Finally, we conclude the paper and suggest
future directions in Section 7.
2. Random Periodic Spectrum Sensing Scheme
In CR networks, a channel usually could be modeled as
an ON-OFF source alternating between ON (busy) and
OFF (idle) periods depending on PUs’ channel-usage
pattern. The sojourn time of a ON period is used for
transmission of PUs themselves and that of a OFF pe-
riod captures the time period in which the channel can be
utilized by CRs’ transmission without causing any
harmful interference to PUs. The distribution of the so-
journ time in the ON state can assumed to be general,
and so is that in the OFF state. Thus the ON-OFF chan-
nel-usage stochastic process describing the behavior of
the channel occupation can form an alternative renewal
process. A renewal period models a time period in which
the PUs and CRs occupy the channel once alternatively.
Hence there are only busy and idle two possible states in
a renewal period accordingly.
Considering that the sensing period of each channel is
a random variable and sensing errors are possible present
at any moment as well as a back-off mechanism is intro-
duced, in this paper we further subdivide the channel in a
renewal period into five kinds of states: normal busy,
available idle, delay idle, false alarm and miss detection.
Below we will describe each state in detail.
Normal busy and available idle are two kinds of nor-
mal available states. The former denotes that PUs are
being served normally and the latter stands for the chan-
nel are being utilized for the transmission of CRs. When
the channel is in normal available, it is sensed once every
random time interval T, i.e. sensing period, to make sure
whether it is in normal busy or available idle.
As soon as the service of PUs completes, a random
back-off time interval T0 is generated to prevent CRs
from occupying the channel at once, within which the
channel is in delay idle state. The back-off mechanism
can help decrease both the connection cost generated by
switching channel state frequently and the short-term
interference probability induced by non-negligible delay
for relinquishing bands by CRs.
Corresponding to the binary hypotheses test of spec-
trum sensing: B0 (null hypothesis indicating that the
sensed channel is available for CRs) vs. B1 (alternative),
there are two kinds of sensing errors: false alarm (the
overlook of an available channel) due to mistaking B0 for
B1 and miss detection (the mistake of identifying an un-
available channel as an opportunity) due to mistaking B1
for B0. When false alarm or miss detection is present, the
channel will transfer to false alarm or miss detection
state. The presence of sensing errors also has significant
effects on the performance of sensing schemes.
Common to most of the periodic spectrum sensing
schemes, the sensing period selection of random periodic
spectrum sensing scheme has strong impact on the sens-
ing efficiency. In order to analyze the optimal sensing
period effectively, an analytical model for sensing period
optimization is constructed in next section.
3. Analytical Model
Owing to the multiplicity of conditionality and correla-
tion that exists among the various random variable in-

C. L. GUO ET AL.399
volved in the random periodic sensing scheme, the
analysis and performance evaluation, especially the de-
termination of optimal sensing period is usually difficult.
Therefore, a set of simplifying assumptions is to be made
for analytical model to be tractable. We assume the fol-
lowing
 The sojourn time of normal busy and available idle
are continuous random variables and drawn from general
cumulative distribution functions (c.d.f.) represented by
F1(t) and F2(t) respectively. Suppose the probability den-
sity functions (p.d.f.) for Fi(t) (i=1,2) are fi(t), means are
1
i

, and,
0()

.
0
()()1 exp[()]
t
ii i
Ftf xdxxdx  

ii
tdF t
0
t
1


 The sensing period T is also a continuous random
variable and follows an arbitrary distribution with c.d.f.,
p.d.f. and mean are 1()Gt,1()
g
y and 1
1

, respectively,
and ( )] 1
1
11
00
() (
tt
G tgy 

1
y dy) 1dy[exp()ET



1
0()tdG t

.
 Delay idle state can transfer to normal busy or
available idle. It is assumed that if a PU reappears during
T0 he can claim the channel and the delay idle will trans-
fer to normal busy with a constant rate γ. Otherwise, if
the back-off timer expires and no one claims the channel,
delay idle transfers to available idle instead. In this case,
T0 follows an arbitrary distribution, and its c.d.f., p.d.f.
and mean are 2()Gt, 2()
g
y and 1
2

, respectively, and
22 2
00
()( )1[( )]
tt
Gtgy dyexpy dy  

1
2
02
0
() ()ETtdG t 




 We assume that α and β denote false alarm and miss
detection probabilities respectively, and the sojourn
times of false alarm and miss detection are arbitrarily
distributed with c.d.f.s (), 1,2
i. Let () i
ht andi Ht i




be their p.d.f.s and means, and
000
()( )1[( )()
tt
iii ii
H
th zdzexpzdztdHt 





When miss detection occurs the channel transfers to
delay idle, whereas when false alarm occurs, in order to
render mathematical tractability, we also assume that the
channel skips normal occupy and transfers to delay idle
too, for the normal occupy time included is small enough
compared to total time of the long-run channel and can
be neglected.
 We assume that the sensing time is small relative to
distribution parameters12 0 ,,,(),()ET ET

 and (1,2)
ii

 ,
and can be negligible. It is also assumed that channel is
in delay idle initially, and all random variables are mutu-
ally independent.
Consider the stochastic process


(), 0St t, where
S(t) denotes the state of the channel at time t, as following
 0 stands for the channel is in delay idle, and the
channel is occupied neither by PUs nor by CRs.
 (, )ik means that the channel is in normal available
states, where i =0/1 represents that the channel is in
available idle/normal busy and k stands for the times of
the channel has been sensed, k =0,1,2, ··· .
 (2, j) denotes that the sensing error is present, where
j =1/2 means that the system is in false alarm/miss detec-
tion.
It is easy to see that, {S(t), t≥0} is a non-Markovian
stochastic process. In order to make the process Mark-
ovian, we need to incorporate the missing information by
adding “supplementary variables” to the state description.
Hence at time t, let Xi (t) (i =1/2) be the remaining nor-
mal busy/available idle time, Yi (t) (i =1/2) the remaining
sensing/delay time, and Zi (t) (i =1/2) the remaining false
alarm /miss detection time. Formally, the evolution of
the stochastic process describing the dynamic behavior
of the channel can be fully characterized by a generalized
Markov process


(),(), (),()|0
ii i
StXtY tZtt, and the
following state probabilities are defined
1
02
2
(,,){ ()(,),(),
()},0,1;0,1, 2,
(,){ ()0,()}
(,){ ()(2,),()},1,2
ik i
ji
PtxydxPStikxXtx dx
yYt ydyik
PtydyPStyY tydy
PtzdzPStjzZtzdzj


 



 


Notations:
0
1211 22
1211 22
1234 11 22
:()1(),: ()(), ()[1()]/
,,:()(),( )
ˆˆ ˆˆ
ˆˆ
,,:()(),()
,:1,1
,,, :,
st
iii iii
F
tFtfsftedtFs fss
ffgffsffsggs
ffgffffg g
MMMM MM
 
 


 





 

 
 
 

23 14
,,sMsMs

 
12112
2
,,:exp[( )],
exp[() ],
x
xy x
EE EEsxE
sx
 x





exp[ () ]
y
Es

y



The possible states of channels and the transitions
among them are shown in Figure 1.
Figure 1. The state transition model of the generalized
Markov process.

C. L. GUO ET AL. 400
According to Figure 1, a few new performance meas-
ures could be defined. The probability of the channel in
available idle state, normal busy state and delay idle state
are Available Idle Probability, Normal Busy Probability
and Delay Idle Probability, respectively. Sensing Fre-
quency is defined the frequency of the occurrence of
channel sensing when the channel is in state(, ),0,1iki

.
False alarm occurs if and only if the channel transfers
from state(0 to state (2,1); while miss de-
tection occurs if and only if the channel transfers from
state to state (2,2). So False Alarm
Frequency and Miss Detection Frequency are defined
accordingly. Also define, ,
, )(0,1,2)kk
)(0,1, 2)kk
1
(1,
()Rt2()Rt 09()
L
t,1()
L
t2() ,
L
t
and 3()
L
tare the Instantaneous Probability of Available
Idle, Normal Busy, Delay Idle, Instantaneous Frequency
of Sensing, False Alarm and Miss Detection at an arbi-
trary time t, respectively. Then define R1, R2, L0, L1, L2
and L3 are the steady state forms of 1()
R
t2(),
R
t,
0()
L
t,1()
L
t2(),
L
t and 3()
L
t, respectively, e.g.,
11
( )lim
t
R
Rt

 .
In CR networks, each channel will go through one or
all kinds of states in which available idle and normal
busy generate rewards by the using for the transmission
of CRs and PUs, respectively, delay idle and false alarm
waste opportunities, miss detection induces interference
to PUs, and sensing has overhead. It is assumed that the
expected rewards per unit time generated by the channel
are e1 or e2 when the channel is in available idle or nor-
mal busy, respectively. And the expected losses per unit
time induced by the delay of channel is c0, the expected
cost of each sensing is c1, the expected false alarm and
miss detection expenses every time are c2 and c3, respec-
tively. We also assumed that expected total rewards gen-
erated by the channel during (0,t] are R, then
3 2
00
00 0
11
()() ()()
tt t
iij j
ij
RteRtdtcLtdtcLtdt



 
(1)
where and c3 are weighting factors (
). Taking Laplace transform on
both sides of (1), we have
12012
,,,,eeccc
123
ccc
1
e
20 1ec
3 2
00
11
() [()()()]/
iij j
ij
RseR scLscLss




(2)
And the expected rewards per unit time in the steady
state is
2
0
3 2
00
11
lim) /lim)
ts
iij j
ij
RRttsR
eRc Lc L

 




s
(3)
where performance measures Ri (i = 1, 2), L0 and Lj (j = 1,
2, 3) could be expressed as functions of the mean sensing
period E(T) (denoted by
x
) and all of them will be il-
lustrated how to obtain in Section 4. Then taking suitable
value for
x
to make R maximum can bring out the op-
timal sensing period
x

. That is
3 2
00
11
arg max)
iij j
xij
x
eRc Lc L

 

(4)
4. Derivation of Performance Measures
In order to derive performance measures for searching
the optimal sensing period, the steady state probabilities
with exact solutions in closed form should be calculated.
How to calculate them using probability analysis and
supplementary variables method is discussed below.
According to Figure 1, we have the following differ-
ential difference equations
210
()()(,, )00,1,2
k
xvyptxy k
tx



 


 (5)
1110 0
0
()()(,,)(,,)
k
k
x
vy ptxyp txy
tx





 



(6)
111
()()(,, )01,2
k
xvyptxyk
tx



 


 (7)
20
()(,) 0vy pty
ty

 

 



(8)
121
()(,) 0zptz
tz









(9)
222
()(,) 0zptz
tz









1
(10)
The above equations are to be solved under the bound-
ary conditions
0
112
00
0
222
0
(,0)
(,,)( )(,)
() (,)()
k
k
pt
pt x ydxdyzptzdz
zp tzdzt










002 0
0
(,0)()(,)pt vyptydy

 (12) 
0
201
0
(, ,0)
()(,, )1,2
k
k
ptx
xptxydxdy k 




(13)
10 0
0
(,0)(, )pt ptydy 

 (14)
1111
0
(, ,0)( )(,,)1,2
kk
ptxvyptxydxdyk 



(15)
2110
0
0
(,0)( )(, ,)
k
k
pt vyptxydx 


 (16)
221 1
0
0
(,0)( )(,,)
k
k
ptvyptxydxdy 


 (17)
And the initial conditions

C. L. GUO ET AL. 401
0
14 2
1
(, ,)
(1)(1)()( )
()/()0,1,2...
k
x
Psxy
kk
M
Mf ffgEF
Gy Dsk

 x





0(0,)( )py y


(18) 


(19)
Taking Laplace transforms of Equations (5)–(18) with
respect to t and solving the equations, we can derive
10
11 11
1141141
(, ,)
{[(1)(1)(1)(1)]()()(1)()()}/()
xx
Psxy
M
gffMgfEFxGyMgfEFxGyDs
 
 

 

 
(20)
1
111
11212214111
(, ,)
[(1)(1)(1)()(1)]()()()/()1, 2...
k
k
x
Psxy
fgMffgffMffEFxGyDsk

 




 
(21)
04141241141212
(,)()( )/()
y
PsyMMMMfMMfMMffEGyDs
 

 
(22) 
2111 12411212421211
(,)[()]exp()() /()PszMfMMfgMffMfgM ffgszHz Ds

  

  (23)
2214 21 42 12
(,)() exp()()/()PszMMfgMMffgszHz Ds



  (24)
where
1411311314111314 214114 21
*
1111214122412 1133121134121
4121 1
()(/) (/) (/)(/)
()(){[/()/][()/(
[(
DsMMMMMMMMfMMMMfMMMMff
)/]
M
fMffhsMMMM MMM MMMf
MM


 
 
    
 
 
 

3412224122412132 1
**
4222 11121114214212
)//][/()/ ]}
[()] ()()()
MMMfMM MMMffg
Mfs fMffhsgMMfMMffhsg

 
   
 


 
 
(25)
1
22
1
2
11111 1
11 1111
2
122212122
22
212121112
22
1212 12
ˆ
(0)()/[()/ ]
ˆˆ
[()/] [()/]
ˆ
{()/{[()]/}
ˆ
{[()]/}
{[( )
DD MMf
MfM ff
Mf
Mf
MM




 
 
 
  
 


 
In accordance with the definitions of
and , we obtain
1()Rt
2()Rt
10
0
0
()(, )
k
k
RtPtxdx


 (26)
21
0
0
()(, )
k
k
Rt Ptxdx


 (27)
Taking Laplace transform on both sides of Equations
(26), (27) and using Equations (19)–(25), we get
*
1
*
1414 122
()
()()/()
s
Rs
M
MgMMfgFsDs



 (28)
1
ˆ


 

 
21
121 2
21
221
112
11 121
11
11 2
ˆˆ ˆ
]/ }}
ˆˆˆ ˆ
()(1/)(
ˆˆ ˆ
)(1/)
ˆˆˆ
ˆ
()(1/)
1
ˆ
f
fg
M
fMfff f
Mff g
MfMf fg



 



 
 
 



 

 

(32)
With the same derivation of R1 and R2, we can get
2
2
*
211241
*
1241 1
*
24241 22
() [()
()]()
()
RsMMM gMf
MMgfFs
MgMf gFs()



 
 




 
 
 
(29)
2
21
011 1
*
12
ˆˆ
(
ˆˆ
)()/
LMMfMf
SM ffD G
1




 

 (33)
According to Figure 1, as the channel is in
state (, ),0,1ik i

, it is being sensed. Using state transfer
frequency formula shown in [9], we get
where, D(s) is given by Equation (25). By applying the
limiting theorem of Laplace transform and L’Hospital’s
rule, we get 101
0
0
()()[(, )(, )]
kk
k
LtxP txPtxdx 



 (34)
11 1
0
*
11122
lim()lim()
ˆ
ˆˆ
()
ts
RRtsRs
()/
M
gM fgFD



 


(30) Taking Laplace transform on Equation (34) and using
Equations (19)–(25), we obtain
22 2
0
1112122
**
112212 2
lim()lim()
ˆˆ
ˆˆ [(
ˆ
ˆˆ ()[() ()]/
ts
RRtsRs
)]
M
gMfMgf
FggfFD
 


 

 

 
 
12
2
21
12
114 14
1124 1
124
24 24
() {()
{( )]
()}
()
LsMMg MMfgf
}/()
M
MMgMf
MMgff
M
gMgffDs



 



 


 


(35)

(31)
where Then

C. L. GUO ET AL. 402
2
2
11 1
0
11 21111
11
lim( )lim()
ˆˆ ˆˆ
ˆˆ [
ˆˆˆ
() ]/
ts
LLt sLs
2
ˆ
M
ffgfgMf
MffgD
f

 




 

 
 
(36)
Similarly, we can derive
2
22
21111 1
211
ˆˆ ˆ
ˆ
(
ˆˆˆ
ˆˆ )/
LMf fgMf
1
ˆ
f
f
gMffgD

 
 
 
 

  (37)
22
31 11
ˆˆˆ
ˆ
()LMfgMffg
 
   ˆ /D
)
(38)
Substituting , L0 and by (30),
(31), (33), (36), (37) and (38) into (4), respectively, the
optimal sensing period
(1,2)
i
Ri(1,2,3
j
Lj
x
 is determined by, i.e.,
the distributions of sojourn time of normal busy and
available idle. For the sojourn time of normal busy is
used for transmission of PUs themselves and that of
available idle can be utilized by CRs’ transmission when
PUs have no data to transmit, are usually deter-
mined by the traffic generated by PUs services in practi-
cal networks.
()
i
Ft
()
i
Ft
5. The Applications of Sensing Period
Optimization
This section illustrates the applications of the analytical
model developed here by using examples involving PU
systems with both voice and data traffic.
5.1. Optimization for PUs with Voice Traffic
A typical phone conversation is marked by periods of
active talking/talk spurts (or ON periods) interleaved by
silence/ listening periods (or OFF periods). The duration
of each period is exponentially distributed, i.e., the so-
journ time of normal busy and available idle follow ex-
ponential distributions with probability density functions
() it
ii
f
te



, and means 1
i

are constants.
It is a special case for analytical model mentioned
above in which 1
i
s

can be substitute by. The tran-
sient rate of available idle to normal busy and that of
normal busy to delay idle are thus reduced to con-
stants
1
iz


i

.
5.2. Optimization for PUs with Data Traffic
In the past, exponential distributions are also frequently
employed to model interarrival times of data calls for its
simplicity, but exponential distributions may not be ap-
propriate in modeling data traffic. Taking Email, an im-
portant application that constitutes a high percentage of
internet traffic, as an example, its traffic can also be
characterized by ON/OFF states. During the ON-state an
email could be transmitted or received, and during the
OFF-state a client is writing or reading an email. Ac-
cording to traffic models included in the UMTS Forum
3G traffic and ITU RM.2072, the Pareto distribution,
which is one of popular heavy-tailed distributions, can be
used to close capture the nature of Email traffic for both
ON and OFF state, i.e., the sojourn time of normal busy
and available idle follow Pareto distributions with prob-
ability density functions 1 () ,
i
i
ii
ii
ft t
t





, and means
/( 1)
ii i



, where0
i

is called the shape parameter
and 0
i

is called the scale parameter.
In order to derive performance measures to search the
optimal sensing period, 1
i
i
ii
t




 and means /(1)
ii i



should substitute()
i
f
tand 1
i


in Equations (5)–(18).
6. Numerical Results and Discussions
Through numerical experiments, we examine the impact
of sensing period selection on maximum expected re-
wards of the channel for different traffic types under
various channel parameters in this section.
6.1. Performance Analysis
According to the characteristics of channels, we first set
the channel parameters as the following 0.02,


10.033,


10.2,c3
c
20.1, 0.2,0.1
 
0.5
,12 0.1,ee 02 0.05,cc

, and let us assume that T0 follows the
uniform distribution with parameters 1yand 10y

,
written 0 T~ (Uy 1 ,10)y

, and 10y.
Two traffic types of PUs are considered, i.e., Type I:
voice traffic and Type II: data traffic. In the case of voice
traffic, the sojourn time of normal busy and available
idle follow exponential distributions with10.04,


20.01


; for data traffic, the sojourn time of normal
busy and available idle follow Pareto distributions with
1
0.8, 1 5



and 22 2.5,60



, i.e., means are
10.04


and 20.01


, respectively. Here 11



and
22



are selected for easy comparison.
To validate the feasibility of the proposed analytical
model for sensing period optimization, numerical exam-
ples are carried out for the following three sensing
schemes, i.e.,
1) Scheme 1: Tx

, that is, it is exactly a fixed period
sensing scheme that sensing is performed at once where
Tx

.
2) Scheme 2: T follows the uniform distribution with
parameters x

 and 1x

, written ~( ,1TU xx) .
3) Scheme 3: T follows the exponential distribution
with parameter
x
, written ~(TEPx).

C. L. GUO ET AL.403
With MATLAB, from (4) we can obtain the optimal
sensing periods
x
 and the maximum expected rewards
R of the channel per unit time in the steady state for each
sensing scheme. Figure 2 and Figure 3 illustrate the ex-
pected rewards of the channel for various values of
x
under Type I and Type II traffic types respectively.
For Type I voice traffic, Figure 2 shows that the
maximum expected rewards of the channel is varied ac-
cording to the distribution of sensing period T. The op-
timal sensing periods for each scheme are {6. 5 ,x


and the maximum expected rewards
are . So the optimal sensing scheme
is sensing randomly with sensing period T following the
exponential distribution when
13.2,7.8}
{1R09.8,106.1,114.9}
7.8x under the
above-mentioned channel parameters, for the scheme
obtained is maximum compared to other two
schemes.
114.9R
For Type II data traffic, Figure 3 shows that the opti-
mal sensing periods for each scheme are
{10.7,17.9,11.3}xand the maximum expected rewards
are . From Figure 3, it can easily
find that the optimal sensing scheme is sensing randomly
with sensing period T following the uniform distribution.
One point that deserves mention is that, compared to the
voice traffic, there are significant differences in maxi-
mum expected rewards among the three schemes in the
case of data traffic.
{77.7,113.7,90.2}R
6.2. Analysis of Channel Parameters
In practical, the optimal sensing scheme is different with
different sets of channel parameters. Next, we study ef-
fects of the setting of various channel parameters on the
optimal sensing scheme.
1) Effects of Distribution Parameters: in both voice
Figure 2. expected rewards vs. expected sensing period for
voice traffic.
Figure 3. expected rewards vs. expected sensing period for
data traffic.
Table 1. The Optimal Rewards R Vs. Optimal Mean Sens-
ing Period
x

for Various Distribution Parameters under
12
0.1,ee

 ,
02
0.05cc 13 , 0.2,0.5cc 0.2, 0.1



,
0
T~ (1,10)Uy y

 and 10y

.
Type Ⅰ Type Ⅱ
112 2
12
{/, /
,,}

 
 
T
x

R
x

R
Tx

7.8 108.1 6.5 121.3
~( ,1)TU xx

 10.2 106.3 9.2 86.5
(0.01,0.04
,0.02,0.03
3,0.1) ~()TEPx 11.8 105.5 7.4 94.6
Tx

9.4 110.5 10.779.8
~( ,1)TU xx

11.9 109.8 8.8 99.6
(0.04,0.01
,0.05,0.03
3,0.1) ~()TEPx 10.2 112.3 5.9 128.9
Tx

10.5 97.5 15.1108.2
~( ,1)TU xx

9.3 99.3 17.5132.4
(0.04,0.01
,0.02,0.1,0
.033) ~()TEPx 8.9 101.63.9 86.7
traffic and data traffic, the distribution parameters
12 12
,,,,

 and 12 12 ,,,,


2
0.1
directly reflect
the sojourn time of normal busy, available idle, delay
idle, false alarm and miss detection state, respectively.
The distribution parameters 11 /0.04  ,22 /0.01  
0 (1 0.02,.033,



) show that the sojourn time of
available idle is longest, which means low or moderate
PU traffics are chosen. Other three distribution parame-
ters are chosen for 3 sensing schemes, and the optimal
rewards R vs. the optimal mean sensing period
x

are
shown in Table 1, respectivel a) (112212
/, /,,,
y.

 
)
is (0.01, 0.04, 0he sojourn time of
normal busy is relatively long and less opportunities can
be used by transmission of CRs; b) (112 212 /, /,,,
.02, 0.033, 0.1). T

  )
is (0.04, 0.01, he sojourn time of de-
lay idle is reduced so that more opportunities can be used
by CRs than 1); and c) (112212
/, /,,,
0.05, 0.033, 0.1). T

  ) is (0.04,
0.01 The sojourn time of miss detec-, 0.02, 0.1, 0.033).

C. L. GUO ET AL. 404
Factors: Weighting factors e1,
e2
tion is longer than that of false alarm. From Table 1, we
clearly see that the optimal sensing scheme is different
with regard to different distribution parameters.
2) Effects of Weighting
for voice traffic: 1) fixd e (=0.dio15) an varus
2

1

;
and 2) fixed1

(= 0.2) and vious 2 ar

, and two differet
situations fodata traffic: 1) fixe2
n
r d

(=0.15 ) and
various 1

; and 2) fixed1

(=0.2) and rious2 va

.
As shown in Figure 4( and Figure 4(b), va ying
, c0, c1, c2 and c3 can be treated as indexes regarding the
importance of 1201
,,,
R
RLL and L2. Generally speaking,
the PU applicati GPS, can only endure minor
interference for acceptable Quality of Service (QOS).
For such applications, the cost of interference induced by
miss detection is more important and c3 is obviously
bigger than all other factors as shown in above-chosen
setting 12 0.1,ee02 0. 05,cc10.2,c ( 30.5c
ons, such as

).
Table 2 showptiman
sensing period
s optimal rewards R and the ol mea
x
 for two different network applica-
tions : a) In the case where the PU is not sensitive to in-
terference, the weighting factors c0 and c2 are both larger
than other factors in order to maximize the utilization of
existing opportunities. Here (e1, e2, c0, c1, c2, c3) is set to
(0.1, 0.1, 0.35, 0.05, 0.35, 0.05). b) For energy-constrained
CR networks such as sensors and mobile ad hoc applica-
tions, frequent sensing is undesirable for high energy
overhead, so the weighting factor c1 should be set rela-
tively large and (0.1, 0.1, 0.05,0.5,0.05,0.1) is chosen.
This table demonstrates that the optimal sensing scheme
is also different with regard to different weighting factors
and from the Table 2 one can easily find which scheme
performs best.
a)r 1

(2

) among 0.1 and 1, we observed that the performance
ooice traffic is not sensitive to which sensing scheme
is preferred. However, for data traffic, with the variation
of 1
f v

and2

, the optimal sensing scheme is changing
amog three sensing schemes and the maximum ex-
pected rewards have big different, as shown in Figure 4(c)
and Figure 4(d). The possible reason is that the optimal
sensing period for voice traffic is depending only on a
constant mean value of the transmission time for primary
users and cognitive radios whereas performance of data
traffic are affected by not only mean value but also the
distributions of the time of transmission for primary us-
ers and cognitive radios.
From the above discussions, we suggest that: 1) for
vo
n
his paper dealt with the random sensing period
ice traffic, the maximum expected rewards of three
sensing scheme is close to each other. In real system, a
fixed period sensing scheme is preferred for simplicity;
and 2) however, the situation changes dramatically for
data traffic, the maximum expected rewards of three
sensing scheme are far different. The analytical model
proposed can be used to search the optimal sensing
scheme, and the analysis can be easily extended to that of
any other distribution of sensing period T.
7. Conclusions and Future Work
6.3. Performance Comparison for Different
ccording to the proposed analytical model for sensing
Traffic Types
AT
period optimization, a sensing scheme with a bigger ex-
pected reward performs better. Figure 2, Figure 3, Table
1 and Table 2 show that the optimal sensing scheme is
different for different traffic types. In Figure 4, we show
the expected rewards for different schemes under
0.02,

10.033,

20.1,

12 0.1, 0.1,ee


02
cc 1 3 0.05, 0.2,c c0.5, and two different situations
scheme in CR networks. An efficient generalized
Markov analytical model for sensing period optimiza-
tion was proposed and studied. How our proposed ana-
lytical model can be applied to PU systems with both
voice and data traffic was also discussed. Both numeri-
cal results and analysis for various channel parameters
and traffic types of PUs were obtained and compared.
We found that sensing period does affect the maximum
expected rewards of the channel, and the proposed
analytical model is valid for the analysis of the case
where the sensing period, the transmission time for
primary users and cognitive radios are all following
arbitrary distributions.
In this work, we assume that the sensing period for
ea
Table 2. The optimal rewardsvs. optimal mean sensing R
period
x
 for various weighting factors under0.04,
 
0. 01,  12 0.02, 0.033,0.1,

  0.   2, 0.1  
0
T~ (1,10)Uy y , and 10y.
(e e2,
1, Type I Type II
c0, c1, c2,
c3)
T
x

R
x

R
Tx
ch channel is different, i.e. the sensing period is asyn-
chronous for all channels. The proposed scheme is suit-
able for the scenario where each CR only sense the
channel for its operating. If each CR is responsible for
sensing more than one channel, the intelligence schedule
algorithm of sensing period should be used to negotiate
among CRs because the quiet period of each channel is
also asynchronous. In future, we would like to develop
practical schedule mechanisms or protocols, which deal
10.4 107.6 9.773.9
~( ,x 1)TUx
(0.1,0.1,
11.3 111.4 14.598.2
0.35,0.0
5,0.35,0.
05) ~()TEPx 8.9 105.3 10.3114.6
Tx 7.5 98.8 11.889.2
~( ,x 1)TUx 15.2 95.6 6.2125.5
(0.1,0.1,
0.05,0.5,
0.05,0.1) ~()TEPx 12.6 101.2 4.975.4

C. L. GUO ET AL. 405
(a) (b)
(c) (d)
Figure 4. Performance comp arison of different traffic types under 0.02,


10.033,


20.1,

 0.2,

 0.1


,
12
0.1,ee 021 3 0.05, 0.2,0.5ccc c with (a) 2

=0.15, (b) 1

= 0.2, (c) 2

= 0.15, and (d) 1

= 0.2.
his work was supported by Natural Science Foundation
] Y. C. Liang, W. S. Leon, Y. H. Zeng, et al., “System
Lau , R. S. Cheng, R. D. Murch, et al., “Adap-
tive quiet period control,” IEEE 802.22–06/0082r0, http://
ic spectrum access-dynamic fre-
el transmission in Cognitive Ra-
r sensing in Cognitive Ra-
sing in cognitive radio
with the coordination of channel sensing among CRs, to
make the proposed sensing scheme more effective in real
CR networks.
8. Acknowledgements
T
que
of China under Grant No. 60772110.
9. References
[1
dio
description and operation principles for IEEE 802.22
WRANs,” IEEE 802.22–05/0094r4, http://www.ieee802.
org/22/.
[2] V. K. N.
www.ieee802.org/22/.
[3] W. D. Hu, D. Willkomm, M. Abusubaih, et al., “Cogni-
tive Radios for dynam
ncy hopping communities for efficient IEEE 802.22
operation,” IEEE Communications Magazine, Vol. 45,
No. 5, pp. 88–87, 2007.
[4] T. Shu, S. G. Cui, and M. Krunz, “Medium access control
for Multi-Channel parall
networks,” In proceedings of IEEE GLOBECOM,
San Francisco, pp. 1–5, 2006.
[5] H. Kim and K. G. Shin, “Efficient discovery of spectrum
opportunities with MAC-Laye
dio networks,” IEEE Transactions on Mobile Computing,
Vol. 7, No. 5, pp. 533–545, 2008.
[6] Y. Zhang, C. Y. Feng, and C. L. Guo, “A flexible sensing
period mechanism of spectrum sen

C. L. GUO ET AL. 406
and S. Li, “A spectrum exchange
the mean fail-
Networks,” The Journal of China Universities of Posts and
Telecommunications, Vol. 31 No. 2, pp. 128–131, 2008.
[7] C. L. Guo, Z. M. Zeng, C. Y. Feng, and Z. Q. Liu, “An
asynchronous spectrum sensing period optimization
mec
model and adaptive fuzzy adjustment algorithm,” Journal
of Electronics & Information Technology, Vol. 31, No. 4,
pp. 920–924, 2009.
[8] C. Han, J. Wang,
hanism in cognitive radio contexts,” In proceedings
of IEEE PIMRC, Finland, pp. 1–5, 2006.
[9] D. H. Shi, “A new method for calculating
ure numbers of a repairable system during (0, t),” Acta
Mathematicac Applicatae Sinica, Vol. 8, No. 1, pp. 101–
110, 1985.