1

Energy-Efficient Cooperative Cognitive Relaying
Schemes for Cognitive Radio Networks

arXiv:1406.2255v3 [cs.NI] 26 Oct 2017

Ahmed El Shafie, Student Member, IEEE, Tamer Khattab, Member, IEEE, Amr El-Keyi, Member, IEEE

Abstract—We investigate a cognitive radio network in which
a primary user (PU) may cooperate with a cognitive radio
user (i.e., a secondary user (SU)) for transmissions of its data
packets. The PU is assumed to be a buffered node operating in
a time-slotted fashion where the time is partitioned into equallength slots. We develop two schemes which involve cooperation
between primary and secondary users. To satisfy certain quality
of service (QoS) requirements, users share time slot duration and
channel frequency bandwidth. Moreover, the SU may leverage the
primary feedback message to further increase both its data rate
and satisfy the PU QoS requirements. The proposed cooperative
schemes are designed such that the SU data rate is maximized
under the constraint that the PU average queueing delay is
maintained less than the average queueing delay in case of noncooperative PU. In addition, the proposed schemes guarantee
the stability of the PU queue and maintain the average energy
emitted by the SU below a certain value. The proposed schemes
also provide more robust and potentially continuous service for
SUs compared to the conventional practice in cognitive networks
where SUs transmit in the spectrum holes and silence sessions
of the PUs. We include primary source burstiness, sensing
errors, and feedback decoding errors to the analysis of our
proposed cooperative schemes. The optimization problems are
solved offline and require a simple 2-dimensional grid-based
search over the optimization variables. Numerical results show
the beneficial gains of the cooperative schemes in terms of SU
data rate and PU throughput, average PU queueing delay, and
average PU energy savings.
Index Terms—Cognitive radio, rate, queue stability, optimization problems.

I. I NTRODUCTION
Secondary utilization of the licensed frequency bands can
efficiently improve the spectral density of the under-utilized
licensed spectrum. Cognitive radio (secondary) users are intelligent devices that use cognitive technologies to adapt
with variations, and exploit methodologies of learning and
reasoning to dynamically reconfigure their communication
parameters [2]–[4]. This allows the secondary users (SUs) to
utilize the spectrum whenever it is free to use and with the
maximum possible data rates.
Cooperative diversity is a recently emerging technique for
wireless communications that has gained wide interest [5]–[8]
Part of this paper was published in the IEEE International Conference on
Computing, Networking and Communications (ICNC), 2015 [1].
A. El Shafie is with the University of Texas at Dallas, USA (e-mail:
ahmed.elshafie@utdallas.edu).
T. Khattab is with Electrical Engineering, Qatar University, Doha, Qatar
(email: tkhattab@ieee.org).
A. El-Keyi is with Wireless Intelligent Networks Center (WINC), Nile
University, Giza, Egypt (aelkeyi@nileuniversity.edu.eg).
The work of T. Khattab is supported by Qatar National Research Fund
(QNRF) under grant number NPRP 7-923-2-344. The statements made herein
are solely the responsibility of the authors.

where multiple channels are used to communicate the same
information symbol. Recently, cooperation in cognitive radio
networks, referred to as the cooperative cognitive relaying,
where the SU helps in relaying some of the undelivered
primary user (PU) packets, has got extensive attention [9]–
[16]. In particular, the SU functions as a relay node for the PU
whenever the PU packet cannot be decoded at its destination.
The authors of [9] showed that the maximum achievable rate
can be achieved by simultaneous transmissions of PU and
SU data signals over the same frequency band. The SU data
signals are jointly encoded with PU data signals via dirty-paper
coding techniques. Hence, the SUs know perfectly the PU’s
data. In [10], the authors assumed that the SU decodes-andforwards the undelivered PU packets during the idle periods
of the PU. The SU maximizes its throughput by adjusting its
transmit power level.
A. Related Work
In [12], the authors investigated the scenario of deploying
a dumb relay node in cognitive radio networks to increase
network spectrum efficiency. The relay node aids both the
PU and the SU. The proposed scheme is investigated for a
network consisting of a pair of PUs and a pair of SUs. In
[13], the authors considered a network with one buffered PU
and one buffered SU where the SU is allowed to access the
channel when the PU’s queue is empty. The SU has a relaying
queue to store a fraction of the undelivered PU packets
controlled through an adjustable admittance factor. A priority
of transmission is given to the relayed PU packets over the SU
own packets. The SU aims at minimizing its average queueing
delay subject to a power budget for the relayed primary
packets. In [15], the authors characterized some fundamental
issues for a wireless shared channel composed of one PU
and one SU. The authors considered a general multi-packet
reception model, where concurrent packet transmission could
be correctly decoded at receivers with a certain probability
that is characterized by the system’s parameters (e.g., packet
length, data rates, time slot duration, bandwidth, etc.). The PU
has unconditional channel access, whereas the SU accesses the
channel based on the activity state of the PU, i.e., active or
inactive, during a time slot. The spectrum sensing process is
impractically assumed to be perfect. The SU is assumed to
be capable of relaying the undelivered PU packets as in [13].
If the PU is sensed to be inactive during a time slot, the SU
accesses the channel with probability one, and if the PU is
active, the SU randomly accesses the channel simultaneously
with the PU or attempts to decode the primary packet with

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the complement probability. The maximum stable throughput
region of the network is obtained via optimizing over the
access probability assigned by the SU during the active periods
of the PU.
Releasing portions of primary systems time slot duration
and bandwidth for the SUs has been considered in several
works, e.g., [11], [14], [17]. In [11], the authors proposed a
spectrum leasing scheme in which PUs may lease their owned
bandwidth for a fraction of time to SUs based on decode-andforward (DF) relaying scheme and distributed space-time coding. In [14], the authors proposed a new cooperative cognitive
scheme, where the PU releases portion of its bandwidth to the
SU. The SU utilizes an amplify-and-forward relaying scheme.
It receives the primary data during the first half of the time slot,
then forwards the amplified data during the second half of the
time slot. In [17], the authors considered an SU equipped with
multiple antennas sharing the spectrum with a single-antenna
energy-aware PU, where the PU aims at maximizing its mean
transmitted packets per joule. The users (SU and PU) split the
time slot duration and the total bandwidth to satisfy certain
quality of service (QoS) for the PU that cannot be attained
without cooperation. Both users maintain data buffers and are
assumed to send one data packet per time slot.
B. Contributions
Given the need for shorter transmission times and low latency communications [18]–[20], we develop two cooperative
cognitive schemes which allow the SU to transmit its data
bits simultaneously with the PU under the constraint of short
communication times and the presence of practical sensing and
feedback cost considerations. Under our proposed schemes,
the PU may cooperate with the SU to enhance its QoS,
i.e., to enhance its average queueing delay and maintain its
queue stability. Hence, cooperation is optional for the PUs.
If cooperation is beneficial for the PU, it releases portion of
its bandwidth and time slot duration for the SU. In turn, the
SU incurs portion of its transmit energy to relay the primary
packets. The SU employs a DF relaying scheme. The time
slot is divided into several intervals (or time phases) that
change according to the adopted cooperative scheme, as will
be explained later. In our first proposed cooperative scheme,
the SU blindly forwards what it receives from the PU even if
the primary destination can decode the data packet correctly
at the end of the PU transmission phase. On the other hand,
in our second proposed scheme, the SU forwards what it
receives from the PU if and only if the primary destination
could not decode the PU transmission of the primary packet;
or if the SU considers the feedback message as a negativeacknowledgement from the primary destination.1 However,
as will be explained later, there is a cost for using the
second cooperative scheme which is a reduction in the time
available for transmission data of users due to the presence
of an additional feedback duration. These practical issues are
quantified analytically in this work.
1 In

this paper, the primary feedback channel is assumed to be modeled as
an erasure channel model and can be undecodable at the secondary terminal.
This will be justified in Section VI.

The contributions of this paper are summarized as follows
• We design two cooperative cognitive schemes which
involve cooperation between the PUs and the SUs. The
two schemes differ in terms of time slot structure and
primary feedback mechanism. Both schemes achieve a
significant PU energy savings.
• We consider practical assumptions for the cognitive radio
network. Precisely, unlike most exiting literature, we
consider spectrum sensing errors and primary feedback
reception errors at the SU. Moreover, we consider the
impact of the time durations spent on spectrum sensing
and feedback message transmission on the achievable
data rates. In addition, the PU data burstiness is taken
into consideration.
• We propose two QoS measures for the PU and include
them in the proposed optimization problems as constraints. Specifically, we assume a constraint on the PU
average queueing delay and a constraint on the stability
of the PU queue. Moreover, we consider a practical
energy constraint on the SU average transmit energy. The
optimization problems are stated under such constraints.
This paper is organized as follows. In the next section, we
introduce the system model adopted in this paper. In Section
III, we analyze the PU queue and derive the PU average
queueing delay and PU queue stability condition. Our first
proposed cooperative scheme is explained in Section V. In
Section VI, we describe our second proposed cooperative
scheme. The numerical results are shown in Section VIII. We
finally conclude the paper in Section IX.
II. S YSTEM M ODEL
We consider a wireless network composed of orthogonal
primary channels, where each channel is used by one PU. Each
primary transmitter-receiver pair coexists with one secondary
transmitter-receiver pair. For simplicity, we focus on one
of those orthogonal channels.2 Each orthogonal channel is
composed of one secondary transmitter ‘s’, one primary transmitter ‘p’, one secondary destination ‘sd’ and one primary
destination ‘pd’. The SU is equipped with two antennas: one
antenna for transmission data and the other for data reception
and spectrum sensing. The PU is equipped with a single
antenna. Moreover, the PU has an infinite-length buffer for
storing a fixed-length packets. The arrivals at the PU queue
are independent and identically distributed (i.i.d.) Bernoulli
random variables from one time slot to another with mean
λp ∈ [0, 1] packets per time slot. Thus, the probability of a
data packet arrival at the PU queue in an arbitrary time slot
is λp . A list of the key variables is given in Table I.
A. Channel Model
We assume an interference wireless channel model, where
concurrent transmissions are assumed to be lost data if the
received signal-to-noise-plus-interference-ratio (SINR) is less
2 As argued in the cognitive radio literature, e.g., [9]–[17] and the references
therein, the proposed cooperative cognitive scheme and theoretical development presented in this paper can be generalized to cognitive radio networks
with more PUs and more SUs.

3

TABLE I: List of Key Variables.
Symbol

Description

τs

Spectrum sensing time duration

Symbol

Description

T and W

Time slot (coherence time) duration
and channel total bandwidth, respectively

R̃

Average SU data rate

P◦

Average transmit information power

Qp

Queue at the PU

τf

Feedback message duration

SU transmission data rate under scheme Pℓ

µp,c

(ℓ)

Re

(ℓ)

and Rb

(ℓ)

Average service rate of the PU queue
under scheme Pℓ

when the PU queue is empty and nonempty, respectively
αj,k

2
Channel gain of the j − k link with mean σj,k

PFA

False alarm probability at the SU

PMD

Misdetection probability at the SU

λp

Average arrival rate at the PU’s queue

Average queueing delay at the PU queue under scheme Pℓ

f

Probability that SU decodes

(ℓ)

Dp,c

the PU’s feedback message
Dp,nc

Average queueing delay at the PU queue with no cooperation

µp,nc

Average service rate of the PU queue
with no cooperation

Eℓ

Secondary mean transmit energy under scheme Pℓ

E

Maximum transmit energy by the SU

b

PU packet size in bits

Ti and Wi

Time and bandwidth assigned to
user i ∈ {p, s} under cooperation

than a predefined threshold, or equivalently, if the instantaneous channel gain is lower than a predefined value.3 We
propose a DF relaying technique, where the SU decodes and
then forwards the PU packet. The SU is assumed to be a fullduplex terminal which means that it can receive and transmit
at the same time. To avoid the loopback self-interference
impairments which can significantly reduce the achievable
rates, we assume that the SU cannot transmit and receive over
the same frequency band. However, the SU can transmit data
over a frequency band and receive over the other.
Both SU and PU transmit with a fixed power spectral
density of P◦ Watts/Hz. The total transmit power changes
based on the used bandwidth per transmission. When a node
transmit over a bandwidth of Wj Hz, the average transmit
power is P◦ Wj Watts. Time is slotted and a slot has a
duration of T seconds. Channel coefficient between node
j and node k, denoted by ζj,k , is distributed according to
a circularly symmetric Gaussian random variable, which is
constant over one slot, but changes independently from one
time slot to another. The expected value of the channel gain
2
αj,k = |ζj,k |2 is σj,k
, where | · | denotes the magnitude of
a complex argument. Each receiving signal is perturbed by a
zero-mean additive white Gaussian noise (AWGN) with power
spectral density N◦ Watts/Hz. The outage of a channel (link)
occurs when the transmission rate exceeds the channel rate.
The outage probability between two nodes j and k without and
with the presence of interference from other nodes are denoted
(I)
by Pj,k and Pj,k , respectively. These outage probabilities are
3 This

will be discussed later in Appendix B.

functions of the number of bits in a data packet, the slot
duration, the transmission bandwidth, the transmit powers, and
the average channel gains as detailed in Appendices A and B.
B. Primary Access and Secondary Access Permission
The PU transmits its data whenever it has a packet to send.
That is, it does not have any restrictions on using the spectrum.
Without cooperation, the PU uses the entire time slot duration
and total bandwidth for its own data signal transmissions,
while the SU does not gain any spectrum/channel access even
if the PU’s queue is empty. This is because, in practice,
the SU may erroneously misdetect the primary activity and
hence it may cause harmful interruption on the primary system
operation, e.g., collisions and packets loss, that can cause
sever packet losses and data delays. In case of cooperation,
and based on the proposed cooperative cognitive schemes that
will be explained shortly, the PU will release a portion of its
time slot duration and total bandwidth to the SU. The SU
will then be allowed to use the spectrum. In practice, the SU
may get permission to access the spectrum if it either provides
economic incentives for the PU or performance enhancement
incentives. Similar to [5], [14], [17] and the references therein,
we consider performance enhancement incentives.
III. Q UEUE S TABILITY, PU Q UEUE M ODEL , AND PU
Q UEUEING D ELAY
A. Stability
A queueing system is said to be stable if its size is bounded
all the time. More specifically, let QT denote the length of

4

queue Q at the beginning of time slot T ∈ { 1, 2, 3, . . . }.
Queue Q is said to be stable if
(1)

For the PU queue, we adopt a late-arrival model where a newly
arrived packet to the queue is not served in the arriving time
slot even if the queue is empty.4 Let ATp denote the number of
arrivals to queue Qp in time slot T, and HpT denote the number
of departures from queue Qp in time slot T. The queue length
evolves according to the following form:
(2)
QT+1
= (QTp − HpT )+ + ATp
p
+
where (z) denotes max(z, 0). We assume that departures
occur before arrivals, and the queue size is measured at the
early beginning of the time slot [21].
B. PU Queueing Delay
Let µp = H̃p , where Ṽ denotes the expected value of V,
be a general notation for the mean service rate of the PU
queue. Solving the state balance equations of the Markov chain
modeling the PU queue (Fig. 1), it is straightforward to show
that the probability that the PU queue has m ≥ 1 packets,
denoted by 0 ≤ νm ≤ 1, is given by
!m
ν0 λp µp
ν0 m
νm =
=
η , m = 1, 2, . . . , ∞ (3)
µp λp µp
µp
λ µ

where η = λp µp . Since the sum over all states’ probabilities
p p
P∞
is equal to one, i.e., m=0 νm = 1, the probability of the
PU queue being empty is obtained by solving the following
equation
∞
∞
X
X
1 m
η =1
(4)
ν0 +
νm = ν0 + ν0
µ
m=1
m=1 p
After some mathematical manipulations and simplifications,
ν0 is given by
λp
(5)
ν0 =1 −
µp
The PU queue is stable if µp > λp . Applying Little’s law, the
PU average queueing delay, denoted by Dp , is then given by
∞
1 X
Dp =
mνm
(6)
λp m=0
Using (3), Dp is rewritten as
∞
ν0 X
Dp =
mη m
(7)
λp µp m=1
Substituting with ν0 into Dp , the PU average queueing delay
is then given by
1 − λp
Dp =
(8)
µp − λp
Following are some important remarks. Firstly, the PU average
queueing delay cannot be less than one time slot, which is
attained when the denominator of (8) equals to the numerator.
This condition implies that µp = 1 packets/time slot, i.e., the
minimum of Dp is attained if the service rate of the PU queue
is equal to unity.
4 This queueing model is considered in many papers, see for example, [10],
[15], [21] and the references therein.

0
u0 l m
p p

1
2
3
u1 l m u2 l m u 3
p p
p p

Fig. 1: Markov chain of the PU’s queue. State self-transitions
are omitted for visual clarity.
70

D p [slots]

lim lim Pr{QT < x} = 1

x→∞ T→∞

lp m p

lp m p

lp

60

p

50

p
p

40

p
30

p
p

20

=0.3 [packets/slot]: analyt.
=0.3 [packets/slot]: sim.
=0.2 [packets/slot]: analyt.
=0.2 [packets/slot]: sim.
=0.1 [packets/slot]: analyt.
=0.1 [packets/slot]: sim.

10
0
0.1

0.2

0.3

0.4

0.5

p

0.6

0.7

0.8

0.9

1

[packets/slot]

Fig. 2: PU average queueing delay versus µp for different
values of λp .

To verify the average queueing delay expression and show
the impact of both λp and µp , we plotted the curves in Fig. 2.
As shown in the figure, increasing µp decreases the average
queueing delays. Moreover, the average queueing delay is
increasing with the increase of the data arrival rate λp . As
shown analytically, the minimum average queueing delay is 1
time slot when µp = 1 packets/slot.
Secondly, the primary packets average queueing delay,
Dp , decreases with increasing of the mean service rate of
the PU queue, µp . On the other hand, µp depends on the
channels outage probabilities which, in turn, are functions of
the links’ parameters, packet size, transmission time durations,
occupied bandwidth, and many other parameters as shown in
Appendices A and B.
Hereinafter, when necessary, we append a second subscript
to the used notations to distinguish between the cases of
cooperation (‘c’) and no cooperation (‘nc’). We also append a
new superscript to distinguish between the proposed schemes.
IV. N ON -C OOPERATIVE

AND

C OOPERATIVE U SERS

A. Non-Cooperative Users
Let T denote the time slot duration that a PU is allowed
to transmit data over a total bandwidth of W Hz. Without
cooperation, the time slot is divided into two non-overlapped
phases: a transmission data phase, which takes place over
the time interval [0, T − τf ]; and a feedback phase whose
length is τf seconds, which takes place over the time interval
[T − τf , T ]. The feedback phase is used by the primary destination to notify the primary transmitter about the decodability
status of its packet. If the PU queue is nonempty, the PU
transmits exactly one packet of size b bits to its respective

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Without cooperation, a data packet at the head of the PU
queue is served if the p → pd link is not in outage. Using
the derived results in Appendix A for the channel outage
probability, the mean service rate of the PU queue, denoted
by µp,nc , is given by
b


2 W (T −τf ) − 1
(9)
µp,nc = exp − N◦
2
P◦ σp,pd
It is noteworthy from (9) that increasing the feedback duration,
τf , decreases the service rate of the PU queue. This is
because the time available for transmission data decreases with
increasing τf ; hence, the outage probability increases which
reduces the service rate. Since the PU transmits with a fixed
rate of Rp = W (Tb−τf ) bits per channel use, increasing W
or T decreases the channel outage probability as seen in (9).
However, increasing Rp decreases the throughput since the
number of decoded bits per seconds is decreased. Hence, one
should compute the number of decoded bits per second per
Hz which is given by
b
τf


2 W T (1− T ) − 1
b
µp,nc = exp − N◦
(10)
2
P◦ σp,pd
WT
Letting Rp =

b
WT

, we have

µp,nc = exp



− N◦

2

Rp
τf
)
(1−
T


−1
Rp

(11)
2
P◦ σp,pd
Using the first derivative of µp,nc in (10) with respect to b,
the optimal packet size is


2
P◦ σp,pd
W
N◦
τf
b⋆ = W T (1 − )
(12)
T
ln(2)
where W(·) is Lambert-W (omega) function. From this interesting result, increasing the feedback duration τf will decrease
the packet size. This is expected since the allowed time to
send a data packet will decrease. On the other hand, we can
see that increasing the time slot duration T or the average
P◦ σ2
at the primary destination increases the
receive SNR Np,pd
◦
optimal packet size. This implies that more packet size can be
supported by the communication system. However, increasing
T and W linearly increase the optimal packet size. When T

2.5

PU throughput [bits/sec/Hz]

destination. The PU and primary destination implement an
Automatic Repeat-reQuest (ARQ) error control scheme. The
primary destination uses the cyclic redundancy code (CRC)
bits attached to each packet to ascertain the decodability status
of the received packet. The retransmission process is based on
an acknowledgment/negative-acknowledgement (ACK/NACK)
mechanism, in which short-length packets are broadcasted by
the primary destination to inform the primary transmitter about
its packet reception status. If the PU receives an ACK over the
time interval [T − τf , T ], it removes the data packet stored at
the head of its queue; otherwise, a retransmission of the packet
is generated at the following time slot(s). The ARQ scheme
is untruncated which means that there is no maximum on the
number of retransmissions and an erroneously received packet
is retransmitted until it is decoded correctly at the primary
destination [10], [13], [15], [22].

2

1.5

1

0.5

0
0

1

2

3

4

5

6

7

Fig. 3: PU throughput [bits/sec/Hz] versus Rp [bits/sec/Hz].
is sufficiently longer than τf , this leads to


2
P◦ σp,pd
W
N
◦
b⋆ = W T
(13)
ln(2)
Thus, the number of bits per channel use Rp that maximizes
the throughput (in bits/sec/Hz) is


2
P◦ σp,pd
⋆
W
N
b
◦
R⋆p =
=
(14)
WT
ln(2)
To verify our analytical finding and show the impact of Rp
on the PU throughput [bits/sec/Hz], we plot Fig. 3. As can be
seen from Fig. 3, the PU throughput increases with Rp until a
peak is reached, then the throughput decreases until it reaches
zero. Hence, there is an optimal value for the packet size (b⋆ or
R⋆p for a given T W ) that maximizes the PU throughput. This
value is given by (14). Increasing the average receive SNR
2
P◦ σp,pd
N◦ increases the PU throughput and also increases
the optimal R⋆p . This matches our discussion below (12).
According to (8), and using (9), the PU average queueing
delay in case of non-cooperative PU is given by
1 − λp
1 − λp
=
Dp,nc =


b
µp,nc − λp
W (T −τf )
exp − N◦ 2 P◦ σ2 −1 − λp
p,pd

(15)
with λp < µp,nc which represents the stability condition of the
PU queue when there is no cooperation.
B. Cooperative Users

When the SU is able to assist the PU with relaying a portion
of the primary packets, the PU, in return, may release a portion
of its spectrum to the SU for its own transmission data if
cooperation is beneficial for the PU. In addition to releasing
some bandwidth for the SU, the PU releases a portion of
its time slot duration to the SU to retransmit the primary
packet. If the cooperation is beneficial for the PU, it cooperates
with the SU. If the PU queue is nonempty, the PU releases
Ws ≤ W Hz to the SU for its own data transmission, and
releases Ts seconds of the time slot to the SU for relaying the
primary packets. The used bandwidth for both transmission
and retransmission of the primary packet is Wp = W−Ws Hz

6

with transmission times Tp and Ts , respectively. Throughout
the paper, we use the analogy of subbands to distinguish
between the primary operational frequency subband, Wp , and
the secondary operational frequency subband, Ws .
1) Spectrum Sensing: The SU senses the primary subband,
Wp , for τs seconds from the beginning of the time slot to detect
the possible activities of the PU. If this subband is sensed to be
idle (unutilized by the PU), the SU exploits its availability by
sending some of its data bits. We assume that the SU employs
an energy-detection spectrum-sensing algorithm. Specifically,
the SU collects a number of samples over a time duration τs ≪
T , measures their energy, and then compares the measured
energy to a predefined threshold to make a decision on the
PU activity [23]. Detection reliability and quality depend on
the sensing duration, τs , and can be enhanced by increasing
τs . Specifically, as τs increases, the primary detection becomes
more reliable at the expense of reducing the time available for
secondary transmission over the primary subband if the PU is
actually inactive. This is the essence of the sensing-throughput
tradeoff in cognitive radio systems [23].
Since the sensing outcome is imperfect and subject to
errors due to AWGN, the SU may interfere with the PU
and cause some packet loss and collisions. To capture the
impact of sensing errors, we define PMD as the probability of
misdetecting the primary activity by the secondary terminal,
which represents the probability of considering the PU inactive
while it is actually active; and PFA as the probability that the
sensor of the secondary terminal generates a false alarm, which
represents the probability of considering the PU active while it
is actually inactive. The values of sensing errors probabilities
are derived in Appendix C.
2) Important Notes and Remarks: In the following, we
state some important notes regarding our proposed cooperative
schemes.
• A communication link is assumed to be ‘ON’ in a given
time slot if it is not in outage. In particular, a link is ON
if the instantaneous data rate of that link is higher than
the used transmission data rate at the transmitter. In this
case, the probability of bit-error rate is very low and can
be neglected. Otherwise, the communication link is said
to be ‘OFF’ (i.e., unable to support the transmission rate).
In other words, the bit-error rate is unbounded (average
symbol error rate is almost 1) and data retransmission
should take place in the following transmission times.
• The CSI of the s → pd, p → s and s → sd links are
assumed to be known accurately at the SU (a similar assumption of knowing the CSI at the transmitters is found
in many papers, for example, [14] and the references
therein).5 This allows the SU to better utilize the spectrum
and helps the PU whenever necessary and possible.
• We assume that the SU always has data bits to transmit and it transmits its data with the instantaneous
channel rate of its link, i.e., s → sd link. This is realized through the implementation of adaptive modulation
5 Note that the channel coefficient between the SU and the primary destination can be estimated by the primary destination and fed back to the SU.
The primary destination only needs to send the state of the channel, i.e., ON
or OFF, which can be realized through a one-bit binary feedback pilot signal.

•

•

•

•

•

•

•

•

schemes which is one of the main advantage of the
cognitive radio devices [14].
Since the SU has the CSI of all the communication links
as explained in the previous bullet, in each time slot, the
SU ascertains the state of the s → pd link, i.e., ON or
OFF link, by comparing αs,pd to the decoding threshold
αth,s,pd. Further details on a link state is provided in
Appendix A. After that, the SU can take decisions based
on the other links to better help the PU.
Since the SU operation is based on the spectrum sensing
outcomes, the time assigned to channel sensing, denoted
by τs , should be less than the PU transmission time Tp
(i.e., τs < Tp ). In particular, the SU cannot set τs to be
longer than the time assigned to PU transmission.
If the p → s link is in outage (i.e., OFF), this means
that the SU will not be able to decode the PU packet
since the noise signal dominates the data signal and the
transmission data rate is higher than the channel rate.
Each PU packet comes with a CRC so that receivers
(primary destination and SU) check the checksum to
indicate the status of the decoded packet. Hence, if the SU
cannot decode the primary packet in a time slot, i.e., the
p → s link is in outage, or if the PU’s queue is empty, the
SU will not waste energy in forwarding what it receives
from the wireless channel because it knows with certainty
that the received packet is a noisy packet (i.e., has no
data when the PU queue is empty). Consequently, the SU
saves its energy from being wasted in a useless primary
data retransmission, and it instead exploits that amount
of energy for the transmission of its own data. This is
critical since the SU energy is constrained and needs to
be optimized.
The data signals transmitted over subband Ws are independent of the data signals transmitted over subband Wp .
Hence, when there is an interference over subband Wp
due to simultaneous transmissions from the SU and the
PU, the data signals over subband Ws do not get affected.
If the PU is active in a given time slot and the SU
misdetects its activity, a concurrent transmission takes
place over the primary subband, Wp . Hence, the SU data
bits transmitted over Wp are lost since the transmission
data rate is higher than the link rate, and the primary
packet could survive if the received SINR is higher than
the decoding threshold. This event occurs with probability
(I)
Pp,pd.6 See Appendix B for further details.
We assume that the primary ARQ feedback is unencrypted and is available to the SU. A similar assumption is found in many references, e.g., [10] and the
references therein.
If the SU transmits concurrently with the primary destination during the feedback phase, the feedback message
(packet) may be undecodable at the PU. For this reason,
the SU remains silent/idle during the primary feedback
duration to avoid disturbing the primary system operation.

6 Throughout

this paper, X = 1−X .

7

Feedback duration
Sensing duration

߬௦

ܶ௣

ܶ௦

߬௙

ܶ

Fig. 4: Time slot structure under proposed scheme P1 . In the
figure, τs is the spectrum sensing time duration, Tp is the
PU transmission time of the primary data packet, Ts is the
time duration assigned to the secondary transmission of the
primary packet, and τf is the feedback duration. Note that
Tp + Ts + τf = T .
V. F IRST P ROPOSED S CHEME
In this section, we explain our first proposed cooperative
scheme, denoted by P1 , and derive the achievable data rates
and the energy emitted by the SU. The time slot structure
under P1 is shown in Fig. 4. In our first proposed cooperative
scheme, the operation of the SU during any arbitrary time slot
changes over four phases: [0, τs ], [τs , Tp ], [Tp , Tp + Ts ], and
[Tp + Ts , Tp + Ts + τf ] (or simply [T − τf , T ]).
A. Scheme Description
Before proceeding to the scheme description, we note that
if the PU is active during a time slot, its transmission takes
place over [0, Tp ], whereas the secondary retransmission of the
primary packet takes place over [Tp , Tp + Ts ]. The operation
of the SU during each phase is described as follows.
1) Time interval [0, τs ]: The SU simultaneously senses the
primary subband, Wp , and transmits its own data over Ws .
The sensing outcome is then used for the secondary operation
over [τs , Tp ].
2) Time interval [τs , Tp ]: If the SU detects the PU to be
active, it simultaneously transmits its own data over Ws , and
attempts to decode the PU transmission over Wp . If the SU
detects the PU to be inactive, it transmits its own data over
both subbands, Wp and Ws . If the PU is active and the SU
finds the primary subband to be free of the PU transmission,
there will be interference between the PU and the SU over
Wp .
3) Time interval [Tp , Tp + Ts ]: If the PU’s queue is empty,
the SU transmits its own data over both subbands. If the links
p → s and s → pd are simultaneously ON and the PU queue is
nonempty, the SU simultaneously transmits its own data over
Ws and retransmits the primary packet over Wp . If either the
p → s link or the s → pd link is OFF, the SU transmits its
own data over both subbands.
4) Time interval [T − τf , T ]: If the PU was active during
[0, Tp ], then its respective receiver broadcasts a feedback
message to indicate the status of the packet decodability.

Hence, the SU transmits its own data over Ws and remains
silent over Wp to avoid causing any interference or disturbance
for the feedback message transmission. If the PU was inactive
during [0, Tp ], there is no feedback message in the current time
slot. However, since the SU does not know the exact state of
the PU during a time slot, it remains idle.
To summarize, the SU does not access the spectrum allocated to the PU, Wp , during the feedback duration to avoid
disturbing the feedback message transmission.
B. PU and SU Data Rates and SU Emitted Energy
(1)

A packet at the head of the PU queue Qp,c is served if
the SU detects the primary activity correctly and either the
direct path or the relaying path7 is not in outage; or if the SU
misdetects the primary activity and the p → pd link is not in
(ℓ)
outage. Let µp,c denote the mean service rate of the PU under
scheme Pℓ , ℓ ∈ {1, 2}. The mean service rate of the PU queue
under scheme P1 is then given by
!
!


(I)
(1)
µp,c = PMD 1 − Pp,pd 1 − Pp,s Ps,pd +PMD 1 − Pp,pd
(16)

(I)
Pp,pd)

where PMD (1 −
denotes the probability of correct
primary packet decoding at the primary destination when the
SU misdetects the primary activity over Wp .
(ℓ)
(ℓ)
Let Re and Rb denote the SU transmission data rate
under scheme Pℓ when the PU
 queue isempty and nonempty,
α
P◦
denote the instantarespectively, and R = log2 1+ s,sd
N◦
neous data rate of the s → sd link in bits/sec/Hz.
Based on the description of scheme P1 , the SU transmission
data rate when the PU queue is empty is given by


R(1)
=
τ
δ
+(T
−τ
)(P
δ
+P
(17)
)+
T
s s
p
s
FA s
FA
s WR
e

where δs = Ws /W . When the PU queue is nonempty, the SU
transmission data rate is given by


(1)
Rb = Tp δs + PMD +PMD Pp,s Ts

+PMD Pp,s Ts (Ps,pd δs +Ps,pd) W R
(18)

(1)
Rb

The term Pp,s appears in
because the SU, when the
p → s link is in outage, uses the entire bandwidth for its own
data transmission. Furthermore, the term Ps,pd appears in the
(1)
expression of Rb because the SU, in each time slot, knows
the channel state between itself and the primary destination
and uses the allocated bandwidth to the PU for its own
transmission data when that channel is in outage.
Let I[L] denote the indicator function, where I[L] = 1 if
the argument is true. The SU transmission data rate when it
operates under scheme Pℓ is then given by
(ℓ)

(ℓ)
(ℓ)
(ℓ)
R(ℓ)
s = I[Qp,c = 0]Re + I[Qp,c 6= 0]Rb

(19)

7 The relaying path is defined as the path connecting the PU to primary
destination through the SU; namely, links p → s and s → pd. Since the
channels are independent, the probability of the relaying path being not in
outage is Pp,s Ps,pd .

8

The expected value of I[L] is equal to the probability of the
argument event. That is,
Ĩ[L] = Pr{L}

(20)

The mean SU transmission data rate is then given by
(ℓ)

(ℓ)
(ℓ)
(ℓ)
R̃(ℓ)
s = Pr{Qp,c = 0}R̃e + Pr{Qp,c 6= 0}R̃b
(ℓ)
Pr{Qp,c

(ℓ)
ν0,c

(21)

(ℓ)
Pr{Qp,c

6= 0} =
= 0} =
and
Recalling that
(ℓ)
1−ν0,c , the mean SU transmission data rate under scheme P1
is then given by


(1)
(1)
R̃s = ν0,c τs δs +(Tp −τs )(PFA δs +PFA)+Ts W Gs


(1)
+ν0,c Tp δs + PMD +PMD Pp,s Ts

+PMD Pp,s Ts (Ps,pd δs +Ps,pd) W Gs
P◦
),
αs,sd N
◦

(22)
which

where Gs is the expected value of log2 (1 +
is given by (see Appendix D for details)
!
!
1
1
1
exp P◦ 2
Gs =
Γ 0, P◦ 2
(23)
ln(2)
N◦ σs,sd
N◦ σs,sd
R∞
where Γ (m, s) = 1/s exp(−z)z m−1dz is the upper incomplete Gamma function.
According to the described scheme, the mean SU transmit
energy, denoted by E1 , is given by


(1)
E1 = ν0,c τs δs +(Tp −τs )(PFA δs +PFA )+Ts W P◦


(24)
(1)
+ν0,c τs δs +(Tp −τs )(PMD δs +PMD )+Ts W P◦
Note that we assume that the maximum average emitted
secondary energy is E; hence, E1 must be at most E.
VI. S ECOND P ROPOSED

SCHEME

In our second scheme, denoted by P2 , we assume a variation
in the primary feedback mechanism to further improve the
achievable performance for both PU and SU. More specifically, we assume the existence of two primary feedback phases
within each time slot. Each transmission of the primary packet
by either the PU or the SU is followed by a feedback phase
to inform the transmitter (PU or SU) about the decodability of
the transmitted packet. In other words, a feedback message is
sent by the primary destination when it receives a copy of the
expected primary packet.8 The first feedback phase is preceded
by the PU transmission of the primary packet, whereas the
second feedback phase is preceded by the SU transmission
of the primary packet. The PU queue drops the packet if it
receives at least one ACK in any time slot. Otherwise, the
packet will be retransmitted by the PU in the following time
slots until its correct decoding at the primary destination.
On the one hand, the gain of this cooperative scheme
over the first proposed scheme lies in its ability to prevent
unnecessary retransmissions of a successfully decoded primary
8 Each packet comes with an identifier (ID) and a certain labeled number
that is generated by the transmitter. In addition, the destination sends the
expected number of the next packet as part of the feedback message.

packet at the primary destination. More specifically, if the
primary destination can decode the PU transmission correctly,
then the SU does not need to retransmit the same primary
packet over the primary subband and over the time assigned
for relaying; hence, the SU can instead use the time assigned
for relaying and the primary subband to transmit its own data
bits to its destination.9 Consequently, using scheme P2 enables
the SU to increase its average transmission rate via using the
allocated bandwidth and time duration for PU transmissions
and its transmit energy to send its own data. On the other
hand, there is a considerable cost due to appending an extra
feedback duration to the time slot. This cost is converted to
an increase in the outage probabilities of the links and the
reduction in the users’ rates. This is because the total time
allocated for data bits and packets transmissions is reduced
by τf seconds relative to the total transmission time in case
of scheme P1 .
Under cooperative scheme P2 , the secondary operation in
any arbitrary time slot changes over five phases as shown in
Fig. 5: [0, τs ], [τs , Tp ], [Tp , Tp + τf ], [Tp +τf , Tp +τf +Ts ] and
[T −τf , T ].
A. Decoding of Primary Feedback Message at the SU
The correctness of the feedback message decoding at the SU
is ascertained using the checksum appended to the feedback
message packet. The decoding of a primary feedback message
at the SU can be modeled as an erasure channel model.
In particular, the primary feedback message is assumed to
be decoded correctly at the SU with probability f . If the
SU cannot decode the primary feedback message in a given
time slot,10 it considers this feedback message as a NACK
feedback message. Another possibility is to assume that the SU
considers the “nothing” as a NACK message with probability
ω and considers it as an ACK message with probability ω.
Using such parameter allows the SU to use a fraction of the
“nothing” events that would be an ACK, which means that the
SU does not need to retransmit the primary packet, for its own
data bits transmission. The SU can optimize over ω to alleviate
wasting the channel resources without further contribution to
the primary service rate when the primary packet is already
decoded successfully at the primary destination. The primary
mean service rate in this case is given by
!
!


(I)
(2)
µp,c = PMD 1−Pp,pd 1−βPp,s Ps,pd +PMD 1−Pp,pd
(25)
where β = f + f ω is the probability of considering the
overheard feedback message as a NACK when the primary
destination sends a NACK feedback (which occurs if the
p → pd link is in outage). From (25), the primary mean
service rate is parameterized by ω. The maximum primary
9 This is because the retransmission of the primary packet by the secondary
transmitter does not provide further contribution to the primary throughput.
In addition, the retransmission of the primary packet causes both energy and
bandwidth losses that can be used otherwise for the SU data transmission.
10 This event is referred to as “nothing” event. The “nothing” event is
considered when the SU fails in decoding the feedback message, or when
the PU is idle at this time slot, i.e., Qp = 0.

9

service rate is attained when ω = 1 since the SU will relay
more PU packets. For simplicity, we consider the case of ω = 1
which guarantees the highest QoS for the PU.

Feedback duration
Sensing duration

B. Scheme Description
The PU transmission occurs over [0, Tp] and the secondary
retransmission of a primary packet occurs over [Tp +τf , Tp +
τf +Ts ]. Note that the feedback message is considered by the
SU as a NACK feedback message 1) if the p → pd link is in
outage and the feedback message is decoded correctly at the
SU terminal; or 2) if the feedback message is undecodable at
the SU. The probability that the SU considers the overheard
primary feedback message as a NACK is then given by
Γf = Pp,pdf +f

(26)

߬௦

ܶ௣

߬௙

ܶ௦

߬௙

ܶ

Fig. 5: Time slot structure under proposed scheme P2 . In this
scheme, there are two feedback message durations. Hence,
Tp + Ts + 2τf = T .

In the sequel of this subsection, we describe the behavior of
the SU during each phase.
1) Time interval [0, τs ] and [τs , Tp ]: The operation of the the SU detects the primary activity correctly, and the direct link
system over the time intervals [0, τs ] and [τs , Tp ] is similar to is in outage, the SU considers the primary feedback message
the first cooperative scheme during the same time intervals.
as a NACK signal, and the relaying link is not in outage; or 3)
2) Time interval [Tp , Tp +τf ]: If the PU queue is nonempty if the SU misdetects the primary activity, and the direct link
during the ongoing time slot, at the end of the PU dedicated is not in outage. The mean service rate of the PU queue is
transmission time, the SU transmits its own data over Ws , similar to the first scheme and is given by
!
!
and remains silent over Wp to avoid causing a concurrent


(I)
(2)
transmission with the feedback message transmitted from the µp = PMD 1−Pp,pd 1−Pp,s Ps,pd +PMD 1 − Pp,pd
primary destination to the PU. If the PU queue is empty during
(27)
the ongoing time slot, the SU transmits its own data over both
We
note
that
the
expression
(27)
is
similar
to
(16).
However,
subbands.
3) Time interval [Tp + τf , T − τf ]: Upon decoding the the maximum assigned transmission data times for users under
entire primary packet, the SU discerns the actual (true) state P2 are lower than P1 as P2 has two feedback durations.
When the PU is inactive, the SU instantaneous transmission
of the PU, i.e., active or inactive. The SU transmits its own
data over both subbands 1) if the PU was active during the rate is givenby


time interval [0, Tp ], the primary destination correctly decoded
R(2)
(28)
e = τs δs +(Tp −τs ) PFA +PFA δs +Ts W R
the PU packet, and the SU successfully decoded the primary
When the PU is active, the SU instantaneous transmission
feedback message, i.e., considered it as an ACK feedback; or
rate is given by
2) if the s → pd link is in outage; or 3) if the PU was inactive
during the time interval [0, Tp ]. If the PU was active during
(2)
Rb = Tp δs +PMD Ts
the time interval [0, Tp], the secondary terminal considered the
!
feedback message sent over [Tp , Tp +τf ] as a NACK feedback,




and the s → pd link is not in outage; the SU simultaneously
× Pp,s Γf Ps,pd δs +Ps,pd +Γf +Pp,s +PMDTs W R
transmits its own data over Ws and retransmits the primary
packet over Wp .
(29)
4) Time interval [T − τf , T ]: If the SU retransmitted the The mean SU transmission data rate is then given by
!
packet over [Tp + τf , T − τf ], another feedback message will

(2)
(2)
be sent over this phase by the primary destination. Hence, the R̃s = ν0,c τs δs +(Tp −τs ) PFA +PFA δs +Ts W Gs
SU simultaneously transmits its own data over Ws and remains
silent over Wp . If the SU decides not to retransmit the primary
(2)
+ν0,c Tp δs +PMD Ts
packet, there will be no primary feedback message. Therefore,
the SU transmits its own data over both subbands. If the PU
!




queue is empty during the ongoing time slot, the SU transmits
× Pp,s Γf Ps,pd δs +Ps,pd +Γf +Pp,s +PMDTs W Gs
its own data over both subbands over this feedback duration
(i.e., [T − τf , T ]).
(30)
C. PU and SU Data Rates and the SU Emitted Energy
(2)

A data packet stored at the head of the PU queue Qp,c is
served in a given time slot 1) if the SU detects the primary
activity correctly, and the direct link is not in outage; or 2) if

According to the description of scheme P2 , the mean SU
transmit energy is given by



(2)
E2 = ν0,c τs δs +(Tp −τs ) PFA +PFA δs +Ts W P◦


(31)
(2)
+ ν0,c τs δs +(Tp −τs )(PMD δs +PMD )+Ts W P◦

10

VII. P ROBLEM F ORMULATION AND P RIMARY M EAN
E NERGY S AVINGS
A. Problem Formulation

max .
s.t.

R̃(ℓ)
s

(ℓ)
Dp,c
< Dp,nc , µ(ℓ)
p,c > λp , 0 ≤ Eℓ ≤ E

1:
2:
3:

We assume that users optimize over Tp = T − τf − Ts and
Wp = W − Ws . It is noteworthy that there is a possibility
to optimize over the spectrum sensing time τs , however, for
simplicity, we assume that the spectrum sensing time is fixed
and predetermined. Sensing time optimization is out of scope
of this paper. The optimization problem is formulated such that
the secondary average data rate is maximized under a certain
PU average queueing delay, the PU queue stability condition,
and an energy constraint on the secondary average transmit
energy, given by Eℓ ≤ E (where E denotes the maximum
average SU transmit energy). The optimization problem under
proposed scheme Pℓ ∈ {P1 , P2 } is stated as follows
Tp ,Wp

Algorithm 1 Optimization Procedure

4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:

Select a large number K
Set i = 1
loop1:
Generate 0 ≤ δp ≤ 1 where δp = Wp /W
loop2:
Set j = 1
(ℓ)
Generate τTs ≤ ∆p ≤ TT where ∆p = Tp /T
Compute Ws = W − Tp and Ts = T (ℓ) − Tp − τs
(ℓ)
Compute Z(i, j) = R̃s in (32)
Set j = j + 1
If j 6= K, goto loop2
Set i = i + 1
If i 6= K, goto loop1
(ℓ)
Select Wp = δp W and Tp = ∆p T that maximize R̃s
(i and j corresponding to highest Z(i, j)) and satisfy the
constraints in (32)

(32)

τs ≤ Tp ≤ T (ℓ) , 0 ≤ Wp ≤ W, Tp +Ts = T (ℓ)
where T (ℓ) is the operational constraint on Tp+Ts when users
(ℓ)
(ℓ)
operate under scheme Pℓ , and Dp,c = (1 − λp )/(µp,c − λp ) is
the average queueing delay of the PU queue under cooperation.
Under our first cooperative scheme, the maximum allowable
transmission time is T−τf ; hence, T (1) = T−τf . On the other
hand, under our second cooperative scheme, the maximum
allowable transmission time is T−2τf ; hence, T (2) = T− 2τf .
It should be pointed out here that if the primary feedback
message is always undecodable at the SU, i.e., f = 1 or if the
p → pd link is always in outage, scheme P1 always outperforms scheme P2 . This is reasonable since the SU will always
retransmit the primary packet with a lower transmission time
for each user due to the existence of two feedback durations
in P2 . In addition, when τf increases, P1 may outperform P2
for some system parameters because it may be the case that
the reduction in the maximum allowable transmission time due
to the presence of an additional feedback duration is higher
than the gain of knowing the status of the primary packet
decodability at the SU before the secondary retransmission of
the primary packet.
The optimization problem (32) is solved numerically using
a two-dimensional grid-based search over Tp and Wp . The
optimal parameters obtained via solving the optimization
problem (32) are announced to both users so that Wp and
Tp are known at the PU and the SU before actual operation
of the communications system. If the optimization problem
is infeasible due to the dissatisfaction of one or more of the
constraints in (32), the SU will not be allowed to use the
spectrum and its achievable rate is zero. A simple method to
solve the optimization problem (32) is to divide the domains
of Tp and Wp into K points. Then, solve the optimization
problem (32) for K 2 times and select the solution that satisfies
the constraints and has the highest objective function. Our
proposed solution to the optimization problem in (32) is stated
in Algorithm 1.
It is worth noting that the PU average queueing delay
constraint can be replaced by a constraint on the mean service

rate of the PU queue. Since the delay constraint is given by
(ℓ)
(ℓ)
Dp,c = (1 − λp )/(µp,c − λp ) < Dp,nc = (1 − λp )/(µp,nc − λp ),
the mean service rate of the PU queue under cooperation must
be greater than the mean service rate of the PU queue without
cooperation. In particular,
µ(ℓ)
p,c > µp,nc

(33)

Combining the delay constraint with the stability constraint,
the PU queue mean service rate should be at least
n
o
µ(ℓ)
(34)
p,c > max µp,nc , λp
B. Mean Primary Energy Savings
In the absence of cooperation, the PU transmission takes
place over T − τf seconds and occupies W Hz. Hence,
the PU energy consumption per time slot is P◦ W (T − τf )
joules/slot. However, when the SU helps the PU in relaying its
packets, the PU transmits only in a fraction Tp /T of the time
slot with transmission bandwidth Wp Hz. Hence, its energy
consumption per time slot is only P◦ Wp Tp ≤ P◦ W (T −τf )
joules/slot. In this case, the average rate of the PU energy
savings, defined as the ratio of the energy savings over the
original energy consumption, is given by
(ℓ)

P◦ W (T −τf )Pr{Qp,nc 6= 0} −P◦ Wp Tp Pr{Qp,c 6= 0}
P◦ W (T −τf )Pr{Qp,nc 6= 0}
(35)
Using the fact that Pr{Qp,nc 6= 0} = λp /µp,nc if λp < µp,nc ,
(ℓ)
(ℓ)
and 1 otherwise, Pr{Qp,c 6= 0} = λp /µp,c , and noting that
there is no cooperation if the PU queue is unstable, we get
Wp Tp max{µp,nc , λp }
φ=1 −
(36)
(ℓ)
W (T −τf )
µp,c
From the above ratio, we can see that the less the bandwidth
and the transmission time that the PU occupies, the more
energy savings for the PU. We note that the PU queue under cooperation should be stable, otherwise, the optimization
problem is infeasible and there will be no cooperation. We also
note that using less bandwidth and shorter transmission time
φ=

11

RESULTS

In this section, we present some simulations of the proposed
cooperative schemes. We define a set of common parameters:
the targeted false alarm probability is PFA = 0.1, W = 10
MHz, T = 5 msec, b = 5000 bits, E = 5 × 10−6 joule,
2
2
2
τs = 0.05T , σs,pd
= σs,sd
= 0.1, σp,s
= 1, P◦ = 10−10 Watts/Hz,
−11
and N◦ = 10
Watts/Hz. Fig. 6 shows the maximum
average SU data rate of our proposed cooperative schemes.
The second proposed scheme is plotted with three different
values of f . The figure reveals the advantage of our second
proposed scheme over our first proposed scheme for f = 0.5
and f = 1. However, for f = 0, the first proposed scheme
outperforms the second one. This is reasonable since when
f = 0 there is no gain from having a feedback message after
the PU transmission; hence, using the second proposed scheme
wastes τf seconds of the time slot that can be used otherwise
in increasing users’ data rates. The figure also demonstrates
the impact of parameter f on the performance of the second
proposed scheme, i.e., scheme P2 . As shown in the figure,
increasing f enhances the performance of scheme P2 . In
addition to the common parameters, the figure is generated
2
= 0.05, τf = 0.05T and the values of f in the
using σp,pd
figure’s legend.
Fig. 7 shows the impact of the feedback message duration,
τf , on the performance of our proposed cooperative schemes.
The mean SU transmission data rate and the PU data arrival
rate feasible range decrease with increasing τf . When the
value of τf is considerable, i.e., τf = 0.2T , the first scheme
outperforms the second scheme. This is because the maximum
allowable transmission data time of nodes under scheme P2 in
this case is T −2τf = 0.6T , whereas the maximum allowable
transmission time under scheme P1 is T−τf = 0.8T . For small
values of τf , the second proposed scheme outperforms the
first scheme since the SU can use the time duration assigned
for relaying and the primary subband to transmit its data in
case of correct packet decoding after the PU transmission.
The parameters used to generate the figure are the common
2
parameters, σp,pd
= 0.05, f = 1 and the values of τf in the
plot.
Figs. 8, 9 and 10 present the primary mean service rate,
the PU average queueing delay, and the average PU power
savings, respectively, under our proposed cooperative schemes.
The case of non-cooperative users is also plotted in Figs. 8 and
9 for comparison purposes. The figures demonstrate the gains
of the proposed schemes for the PU over the non-cooperation
case. Note that without cooperation between the two users,
the PU queue is unstable when λp > 0.2 packets/slot and,
hence, the queueing delay is unbounded. On the other hand,
with cooperation, the PU queue remains stable over the range
from λp = 0 to λp = 0.95 packets/slot. The second scheme
achieves better performance than the first scheme in terms of

4

4.5

x 10

4

Mean secondary rate
[bits/slot]

VIII. N UMERICAL

primary QoS. Fig. 10 reveals that more that 95% of the average
primary energy will be saved for λp = 0.2 packets/slot.
When λp = 0.8 packets/slot, the primary energy savings is
almost 78%. For λp ≥ 0.95, the PU queue becomes unstable
even with cooperation; hence, the cooperation becomes nonbeneficial for the PU and the PU ceases cooperation with the
SU. Hence, the SU doest gain any access to the spectrum, and
the primary energy savings becomes zero since the PU will
send its data over the entire time slot duration and channel
bandwidth. The parameters used to generate the figures are the
2
2
common parameters, σp,pd
= 0.005, σs,pd
= 1, τf = 0.05T and
f = 1. Note that the performance of our two proposed schemes
are close to each other because the outage probability of the
primary channel is high and the direct link (i.e., the p → pd
link) is in outage most of the time. Hence, under scheme P2 ,
the SU retransmits the primary packets almost every time slot
instead of transmitting its own data signals. Accordingly, both
proposed schemes almost achieve the same performance.

3.5
3
2.5

P2 , f = 1

2

P2 , f = 0.5
P2 , f = 0

1.5

P1

1
0.5
0
0

0.2

0.4

0.6

λp [packets/slot]

0.8

1

Fig. 6: The maximum SU data rate in bits per slot for the
proposed schemes. Scheme P2 is plotted with three different
values of primary feedback correct decoding, f .

4

4.5

x 10

P1
P2

4

Mean secondary rate
[bits/slot]

improves the low probability of intercept/low probability of
detection (LPD/LPI) characteristics of the communication link
that appears to be especially critical in military applications.
Hence, it is always useful to use shorter transmission times
and lower bandwidth.

τf = 0.05T

3.5
3
2.5
2
1.5

τf = 0.2T

1
0.5
0
0

0.2

0.4

0.6

0.8

1

λp [packets/slot]

Fig. 7: The maximum mean SU data rate in bits per time
slot. The schemes are plotted for two values of the feedback
duration τf .
IX. C ONCLUSIONS
In this paper, we developed two cooperative cognitive
schemes which allow the SU to access the primary spectrum

12

1

Primary mean service rate
[packets/slot]

Average primary energy savings

P1

0.8

P2

0.7

Without cooperation

0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4

0.5

λp [packets/slot]

0.6

0.7

0.8

Primary queueing delay
[time slots]

P1
P2
Without cooperation

100

50

0.05

0.1

0.15

0.85
0.8
0.75
0.7
0.65
0.2

0.4

0.6

λp [packets/slot]

0.8

1

Fig. 10: PU power savings.

be the transmission rate of node j while communicating with
node k, γj,k be the received SINR at node k when node j
communicates with node k, and αj,k be the associated channel
2
gain with mean σj,k
, which is exponentially distributed in the
case of Rayleigh fading. The probability of channel outage
between node j and node k is given by [24]



(37)
Pj,k = Pr rj,k > log2 1 + γj,k

150

0
0

P2

0.9

0

Fig. 8: The maximum mean primary stable throughput for the
proposed schemes. The case of non-cooperative users is also
plotted for comparison.

P1

0.95

0.2

λp [packets/slot]

0.25

0.3

Fig. 9: The PU average queueing delay for the proposed
schemes. The case of non-cooperative SU is also plotted for
comparison purposes.

simultaneously with the PU. We showed the gains of our
proposed cooperative schemes for the SUs and PUs. We also
addressed the impact of the feedback process on users’ data
rates. Each of our proposed schemes can outperform the other
for certain system parameters and they differ in terms of time
slot structure. We showed that as the probability of feedback
message decoding decreases, the second proposed scheme
loses its advantage over the first proposed scheme. The PU
energy savings under cooperation is more than 60% for most
of the PU packet arrival rate. Moreover, at low mean arrival
rate at the PU data queue, the PU energy savings can be
more than 95%. We also showed a significant reduction in the
average queueing delay of the PU queue under cooperation
relative to the no-cooperation case. As a future work, we can
investigate the battery-based system where the communication
nodes are equipped with rechargeable batteries with certain
energy arrival rates.
A PPENDIX A
In this Appendix, we present the outage probability expression of a link when the transmitter communicates with its
respective receiver alone, i.e., without interference. Let rj,k

where Pr{·} denotes the probability of the event in the
P◦ αj,k
argument, and γj,k =
N◦ . The formula (37) can be
rewritten as


N◦  rj,k
2 −1
(38)
Pj,k = Pr αj,k <
P◦
rj,k
◦
Let αth,j,k = N
− 1). We note that if αj,k < αth,j,k ,
P◦ (2
the channel is in outage (OFF), whereas if αj,k ≥ αth,j,k , the
channel is not in outage (ON). It is worth pointing out here
that increasing the transmission data time and the bandwidth
assigned to any of the terminals decrease the outage probability, or equivalently increase the rate, of the link between
that terminal and its respective receiver. That is, the outage
probability of any of the links decreases exponentially with the
increase of the transmission time and the bandwidth assigned
to the transmitting node.
If the SU is available to assist, when the PU’s queue is
nonempty, the PU sends a packet of size b bits over Tp second
and frequency bandwidth Wp . Hence, the PU transmission rate
is given by
rp,pd =

b
Wp Tp

(39)

When the PU communicates with its destination alone, i.e.,
without interference from the SU, the link between the PU
and the primary destination (i.e., the p → pd link) is not in
outage with probability
b


2 Wp Tp − 1
(40)
Pp,pd = exp − N◦
2
P◦ σp,pd

The probability of primary packet correct decoding at the
SU is equal to the probability of the p → s link being not in
outage. This is given by a formula similar to the one in (40)

13

with the relevant parameters of the p → s link. That is,
b


2 Wp Tp − 1
(41)
Pp,s = exp − N◦
2
P◦ σp,s
The SU relays (retransmits) the primary packet over Ts
seconds and frequency bandwidth Wp . Hence, the transmission
rate of the relayed primary packet is given by
b
rs,pd =
(42)
Wp Ts
The relayed primary packet transmitted by the SU is correctly
decoded at the primary destination with probability
b


2 Wp Ts − 1
Ps,pd = exp − N◦
(43)
2
P◦ σs,pd
where (43) is the probability that the s → pd link is not in
outage.
A PPENDIX B
When the SU and the PU transmit at the same time over
the primary subband, the outage event of the p → pd link is
given by



b
αp,pd P◦
(I)
> log2 1 +
Pp,pd = Pr
(44)
Tp Wp
N◦ + αs,pd P◦
This can be written 
as

b
αp,pd P◦
(I)
Tp Wp
−1>
Pp,pd = Pr 2
(45)
N◦ + αs,pd P◦
Since the channels are independent, the region where the
b
αp,pd P◦
is satisfied can be easily
inequality 2 Tp Wp − 1 > N◦ +α
s,pd P◦
obtained. After some algebra, the probability of primary packet
correct decoding when the SU interrupts the PU transmission
data over Wp is given by
(I)

Pp,pd =

1+

Pp,pd
2
b
σs,pd
(2 Wp Tp
2
σp,pd

− 1)

≤ Pp,pd

(46)

From expression (46), the successful transmission in case of
interference is outer bounded by Pp,pd . This quantifies the
reduction of primary throughput due to concurrent transmission which may occur due to sensing errors. As the message
rate increases WpbTp , the outage probability Pp,pd increases.
Under interference, the correct decoding probability decreases
withb the same amount in addition to a reduction factor of
(2 Wp Tp − 1). Moreover, as the cross-channel average gain,
2
given by σs,pd
, decreases, the correct decoding probability
increases since the interference is weak. Actually, what really
matters is the ratio of the average of the direct (main) channel
σ2
σ2
and the interference channel, which is given by σ2s,pd . As σ2s,pd
p,pd
p,pd
decreases, the interference can cause no impact on the correct
2
σ
packet decoding. When σ2s,pd or the transmission rate WpbTp
p,pd

is very small, we have

(I)
Pp,pd

≈ Pp,pd .

of spectrum sensing [23] and Wp is the primary bandwidth in
case of cooperation) is described as follows
H1 : s(k̂) = ζp,s x(k̂) + ε(k̂)
H0 : s(k̂) = ε(k̂)
H(s) =

In this appendix, we derive the sensing errors probabilities
at the SU. The detection problem at time slot T (assuming that
τs Fs is an integer, where Fs = Wp is the sampling frequency

(48)

k̂=1

where |ζp,s |2 = αp,s is channel gain of of the p → s link,
hypotheses H1 and H0 denote the cases where the PT is active
and inactive, respectively, τs Fs is the total number of used
samples for primary activity detection, ε is the noise instantaneous value at time slot T with variance Np = N◦ Wp , x is
the PU transmitted signal at slot T with variance Pp = P◦ Wp ,
x(k̂) is the k̂-th sample of the PU transmit signal, s(k̂) is the
k̂-th received sample of the primary signal at the SU, and H(·)
is the test statistic of the energy detector.
The quality of the sensing process outcome is determined
by the probability of detection, PD = 1 − PMD , and the
probability of false alarm, PFA , which are defined as the
probabilities that the spectrum sensing scheme detects a PU
under hypotheses H1 and H0 , respectively. Using the central
limit theorem (CLT), the test statistic H for hypothesis Hθ ,
θ ∈ {0, 1}, can be approximated by Gaussian distributions
[23] with parameters
(θαp,s Pp + Np )2
(49)
Λθ = θαp,s Pp + Np , σθ2 =
Fs τs
where Λθ and σθ2 denote the mean and the variance of the
Gaussian distribution for the hypothesis Hθ , where θ ∈ {0, 1}.
Since αp,s is Exponentially distributed random variable with
2
parameter 1/σp,s
, the probabilities PFA and PD can be written
as
PD = Pr{H(s) > ǫ|H1 }
N
exp( σ2 pPp ) Z ∞ p
z
ǫ
p,s
)dz
Q( Fs τs [ −1]) exp(− 2
=
2 P
σp,s
z
σp,s Pp
p
Np
(50)
p
ǫ
PFA = Pr{H(s) > ǫ|H0 } = Q( Fs τs [
− 1])
(51)
Np

where exp(·) denotes the exponential
R ∞ function, ǫ is the energy
threshold and Q(Y) = √12π Y exp(−z 2 /2)dz is the Qfunction.
For a targeted false alarm probability, PFA , the value of the
threshold ǫ is given by
 Q−1 (P )

√ FA + 1
(52)
ǫ = Np
Fs τs
Thus, for a targeted false alarm probability, PFA , the probability of misdetection is given by substituting Eqn. (52) into
Eqn. (50). That is,
1
1
exp( 2
Np )
2 P
σp,s
σ
p
p,s Pp
 −1

!
Z ∞ p
Np Q√F(PτFA) +1

z
s s
Fs τs
Q
×
)dz
−1 exp(− 2
z
σ
Np
p,s Pp
(53)

PMD = 1−
A PPENDIX C

Fs τs
1 X
|s(k̂)|2
Fs τs

(47)

14

where Q−1 (·) is the inverse of Q-function.
A PPENDIX D
In this Appendix, we derive the average value of SU
instantaneous data rate R = log2 (1 + αs,sd γs,sd ). It can be
shown that Z
∞
αs,sd
P◦
1
) exp(− 2 ) dαs,sd
Gs = 2
log2 (1 + αs,sd
σs,sd 0
N◦
σs,sd
!
1
1
1
=
exp(
2 )Γ 0, γ
2
ln(2)
γs,sd σs,sd
s,sd σs,sd
(54)
where Γ (·, ·) is the upper incomplete Gamma function.

P◦
. Integration by parts and rearranging
Proof. Let γs,sd = N
◦
the resultant, the expression is given by
!
Z ∞
αs,sd
Gs = −
log2 (1 + αs,sd γs,sd ) d exp − 2
σs,sd
0
!∞
αs,sd
= − log2 (1 + αs,sd γs,sd ) exp − 2
σs,sd
0
|
{z
}
zero
!
Z ∞
γs,sd
1
αs,sd
dαs,sd
+
exp − 2
ln(2) 0
σs,sd 1 + αs,sd γs,sd
(55)
After eliminating the zero term, Gs becomes
!
Z ∞
1
γs,sd
αs,sd
Gs =
dαs,sd (56)
exp − 2
ln(2) 0
σs,sd 1 + αs,sd γs,sd
Letting z = 1 + y, Gs becomes
!
Z ∞
1
1
y
dy
Gs =
exp −
2
ln(2) 0
γs,sd σs,sd 1 + y
!
Z ∞
1
z−1
1
dy
exp −
=
2
ln(2) 1
γs,sd σs,sd z
!Z
!
∞
1
z
1
1
exp
dz
exp −
=
2
2
ln(2)
γs,sd σs,sd
γs,sd σs,sd z
1
(57)
Letting γs,sdzσ2 = q, we get
s,sd
!Z
∞
1
1
1
exp (−q) dq
exp
Gs =
2
1
ln(2)
γs,sd σs,sd
q
γs,sd σ2
s,sd
(58)
!
!
1
1
1
=
Γ 0,
exp
2
2
ln(2)
γs,sd σs,sd
γs,sd σs,sd

where Γ (·, ·) is the upper incomplete Gamma function.
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