2013 IEEE Wireless Communications and Networking Conference (WCNC): MAC

Stability Analysis In a Cognitive Radio System with Cooperative Beamforming

Mohammed Karmooset, Ahmed Sultant, Moustafa Youssef* tDepartment of Electrical Engineering, Alexandria University, Alexandria, Egypt

*Wireless Research Center, Egypt-Japan Univ. of Sc. & Tech. (E-JUST) and Alex. Univ., Alexandria, Egypt

Abstract-We consider a cognitive radio setting in which a relay-assisted secondary link employs cooperative beamforming to enhance its throughput and to provide protection to the primary receiver from interference. We assume the presence of infinite bufl"ers at both the primary and secondary transmitters and characterize the maximum stable throughput region exactly using the dominant system approach. Numerical examples are provided to give insights into the impact of power control on the stability region.

Index Terms-Cognitive radio, cooperative beamforming, re­lay, queue stability

I. INTRODUCTION

Cognitive radio (CR) technology has been proposed as a

potential solution to the problem of spectrum underutilization.

The CR-enabled unlicensed or secondary devices may use

the spectrum if they do not disrupt the operation of the

licensed or primary users. In the opportunistic spectrum access

(OSA) model, a secondary user utilizes the spectrum whenever

a primary user is not, and vacates the spectrum upon the

presence of the primary user. This coexistence strategy ensures

high level of protection to the primary user while allowing the

secondary user to make use of the silent periods of the primary

user.

In this work we investigate buffered primary and secondary

users and, hence, include a queueing analysis of the proposed

model. Such analysis has been considered in the study of

cognitive radio networks (CRNs) under different scenarios.

One of the main objectives is to characterize the achievable

arrival rates of the network users which guarantee stability

of all system queues. For instance, in [1], an overlay CRN

consisting of a primary and secondary pairs is considered,

and maximum secondary stable throughput is characterized for

fixed primary throughput. The model is extended to include the

insertion of a secondary relay to assist the primary transmis­

sion. In [2], the model is further extended to acconunodate

several secondary users which coexist with a primary link

through a collision channel. Secondary users are required to

comply to interference and QoS constraints of the primary

user. However, the major drawback in the assumed model

is that secondary users are only allowed to transmit if the

primary user is not detected in the spectrum. This drastically

reduces the available spectrum opportunities for the secondary

network and therefore limits the achievable service rate of the

secondary network.

We consider in this paper a relay-assisted secondary link.

Relay-based cooperative beamforming has been widely con­

sidered in CRNs as a means to increase the available spectrum

opportunities for single-antenna users [3], [4]. By providing

secondary users with a set of assisting relays, the secondary

user can utilize the spatial diversity of the relays by forming a

virtual antenna array (VAA). Moreover, using the appropriate

beamforming vector, secondary transmission can be nulled out

at the primary receiver allowing secondary operation even

when the primary user is sensed to be active. In [5], [6], a

set of relays equipped with finite-sized buffers are assumed to

help the secondary transmission in a cooperative beamforming

manner. Scheduling is studied to manage the two phases of

secondary transmissions: from secondary source to relays, and

from relays to destination, while minimizing the delay in

secondary user transmission. Spectrum sensing is assumed to

be perfect, thereby rendering the primary receiver completely

protected from interference.

In this paper, we consider buffered users and a relay-assisted

secondary link, but take the spectrum sensing errors into

account. The secondary user is able to use the spectrum if the

spectrum is vacant, or simultaneously with the primary user

by applying beamforming techniques to utilize unused spatial

dimensions. We aim at characterizing the maximum stable

throughput region of the primary and secondary user. Fur­

thermore, we assume that secondary transmission is equipped

with power control capabilities in order to optimize system

performance. Specifically, optimal power allocation is obtained

to maximize the stable region achieved by the primary and

secondary users. In order to arrive at the provided results, we

resort to the concept of "dominant systems" to decouple the

interacting queues and show that it provides the exact maxi­

mum stable throughput region. To the best of our knowledge,

this is the first study of the proposed cognitive setting from

a queueing theory point of view, while avoiding unrealistic

assumptions such as perfect spectrum sensing.

The rest of the paper is organized as follows. Section

II introduces the assumed system model. In Section III we

characterize the mean service rates of both users. In Section

IV we use the concept of dominant systems to perform the

queueing stability analysis. We provide numerical examples

to provide insights into the obtained problem formulation in

Section V, and we conclude the paper in Section VI.

978-1-4673-5939-9/13/$31.00 ©2013 IEEE 637

K Relaying Nodes

Fig. I. The system model consists of a primary link composed of primary channel with gain Hp connecting the primary transmitter (PU-Tx) and primary receiver (PU-Rx). The secondary transmitter (SU-Tx) is linked to the secondary receiver (SU-Rx) via K relays, where Rk is the kth relay. The transmission between SU-Tx and the relays is assumed to be perfect and unaffected by the interference from PU-Tx. Channel Hsk is the channel between Rk and SU-Rx. SU-Tx overhears PU-Tx's transmission and can sense its activity. Channel Hps is the interference channel from PU-Tx to SU-Rx, whereas channel Hspk is the interference channel from Rk to PU-Rx.

II. SYSTEM MODEL

We assume the presence of a single secondary transmitter

trying to communicate with its respective receiver opportunis­

tically in the presence of a primary network that consists of

a transmitter-receiver pair. The transmit power used by the

primary transmitter is Pp . The primary link operates in a time­

slotted fashion. Each of the users considered are equipped with

single antennas. The primary and secondary transmitters are

both equipped with infinite buffers, Qp and Qs, respectively,

to store their data packets. The arrivals at the primary and

secondary queues are independent and identically distributed

(i.i.d) Bernoulli random variables from slot to slot with means

Ap and As, respectively. Arrival processes at the primary and

secondary buffers are statistically independent of one another.

Because of the significant pathloss to the secondary desti­

nation, direct transmission from the secondary transmitter is

undecodable at the destination. Hence, the secondary source­

destination communication is assisted by a relay network

of K relays which receives the packets from the secondary

transmitter and operates in a decode-and-forward fashion.

When the relays operate, their total transmit power is Ps.

Power Ps can be changed in order to control the level of

interference that may be inflicted on the primary receiver, and

such that Ps ::; Pmax. The relays are assumed to be in the

vicinity of the secondary transmitter. This allows low-power

transmission by the secondary transmitter, thereby reducing

significantly the interference inflicted on the primary link.

Moreover, due to the small pathloss from the secondary source

to the relays, the communication between them is almost error­

free.

The thermal noise at each of the receivers in the system

follows a complex Gaussian distribution eN rv (0,1). We

assume that all the channels are i.i.d and follow a complex

Gaussian distribution eN rv (0,1). We adopt a slow fading

model where the channel gains remain constant over tens of

time slots. Channel estimation occurs during a tiny fraction

of the time slots via overhearing the transmissions by the

secondary and primary receivers. The secondary receiver may

transmit dedicated symbols for channel estimation at the

relays, wheres the primary receiver is assumed to transmit

automatic repeat request (ARQ) feedback to the primary trans­

mitter in the form of acknowledgment (ACK) and negative

acknowledgment (NACK) packets. Due to the broadcast nature

of wireless communications, these packets can be overheard

by the relays and their received signal strength can be used

to estimate the gains of the channels between themselves and

the primary receiver. We assume that channel estimation is

perfect and that the relays forward the estimated channels to

the secondary transmitter.]

The primary user transmits the packet at the head of its

queue starting at the beginning of the time slot provided that

its queue is nonempty. The secondary transmitter senses the

channel at the beginning of the time slot in order to determine

the state of primary activity. Based on the sensing outcome,

and given its knowledge of the channel gains between the

relays and the primary and secondary receivers, the secondary

transmitter computes the precoding or beamforming vector.

It sends a data packet over a fraction of the slot duration

to the relays. The beamforming vector is incorporated within

the packet. 2 Due to the assumption of relays' vicinity to the

secondary transmitter, the packet is received correctly by the

relays. There are two possibilities for the beamforming vector

depending on the spectrum sensing outcome.

a) The primary user is sensed to be idle: In this case, the

secondary relays employ conventional transmit beamforming

to enhance the signal-to-interference-plus-noise ratio (SINR)

at the secondary receiver. The beamforming vector, Wa, is

given by:

(1)

where Hs is a column vector of length K representing the

channels between the secondary relays and the secondary

receiver, and 11.11 represents the L2-norm of a vector.

b) The primary user is sensed to be active: In this case,

the secondary transmission is still allowed provided that the

relays use a beamforming vector designed to null out the signal

at the direction of the primary receiver while maximizing

SINR at the secondary receiver. Assume that Hsp is a column

lif the channels vary more frequently, then the reduction in throughput due to the time needed for channel estimation will be considerable.

2The secondary packet is assumed to have a different size from the primary packet to accommodate the fact that it is sent via a two-hop link within a single time slot.

638

vector of length K representing the channels between the

relays and the primary receiver. Thus, the beamforming vector,

wp, is given by:

(2)

where superscript H denotes vector Hermitian, <I> = �;;,�� is the projection matrix onto vector Hsp, and I is the K x K identity matrix.

The spectrum sensing process is not perfect. The probability

of misdetecting the activity of the primary transmitter is given

by Pmd, whereas the false alarm probability, which is the

probability of sensing the primary transmitter to be busy while

it is idle, is given by Pfa'

III. QUEUE SERVICE RATES

Here, we derive the achievable service rates for both primary

and secondary queues.

A. Primary User Service Rate We can now enumerate the possible situations that can occur

in case the primary user has a packet to transmit.

1) Secondary user's queue is empty: We denote the prob­

ability of Qs being empty as Pr{ Qs = O}. In this case, the

channel is used solely by the primary user. Given that the noise

variance is unity, the probability of outage, defined as the event

when the receive SINR falls below a certain threshold, is given

by

Pout,p = Pr{Pp1Hp12 < ,Bp} = 1-exp(-� ) (3) p where Hp is the primary link complex channel gain, and

,Bp is the SINR threshold for correct reception of primary

transmission. The primary service rate in this case is 1-Pout,p' 2) Secondary user's queue is nonempty and the primary

user is detected: This situation happens with a probability

equal to (1 - Pmd)Pr{Qs -I- O}, where Pr{Qs -I- O} = 1 -PI' { Q s = O}. Given that the relays use the beamforming vector

given in (2), the interference caused by secondary transmission

is eliminated at the primary receiver. Therefore, the service rate

is also given by 1 - Pout,p' 3) Secondary user's queue is nonempty and the primary

user is misdetected: This situation happens with a probability

equal to PmdPr{Qs -I- O}. Since the secondary user misdetects

the activity of the primary transmitter, the relays employ

beamforming vector Wa given in (1). We denote the outage

probability in this case by P�utp, where the superscript 'md'

denotes a situation of misdetection with the secondary trans­

mission causing interference at the primary receiver. Outage

probability P�utp is given by:

md _ Pr{ PplHpl2 } Pout,p - IHn,wa12 + 1 <,Bp

The service rate is then equal to 1 - P�utp.

(4)

Based on the preceding enumeration, the primary mean

service rate, J.Lp, can be written as:

J.Lp = (1 - Pout,p) (Pr{Qs = O} + (1- Pmd)Pr{Qs -I- O}) + (1 - P�utp)PmdPr{Qs -I- O}

(5)

B. Secondary User Service Rate We enumerate the possible situations that a secondary

transmitter can find when trying to transmit. 1) Primary queue is empty and the secondary user detects

the channel to be vacant: Denoting the probability of the

primary queue being empty by Pr{Qp = O}, this case happens

with a probability (1 - pja)pr{Qp = O}. The beamforming

vector Wa is used and the outage probability of the secondary

transmission is then equal to

(6)

where ,Bs is the SINR threshold for correct reception of

secondary transmission. 2) Primary queue is empty and the secondary user finds

the channel busy: This case happens with a probability

PfaPr{Qp = O}. In this case, the secondary user is falsely led

to use the beamforming vector of the form (2). The outage

probability of the secondary link in this case, denoted by fa . .

b Pout,s' IS gIven y:

P��t,s = Pr{IH�wpI2 < ,Bs} (7)

3) Primary queue is nonempty and the secondary user detects primary activity: This case happens with a probability

(1- Pmd)Pr{Qp -I- O}, where Pr{Qp -I- O} is the probability

of the primary queue being nonempty. The outage probability

in this case is

(d) _ .{ IH,!iwp12 } Pout,s - PI PplHpsl2 + 1 <,Bs . (8)

where Hps is the complex gain of the channel between the

primary transmitter and the secondary receiver. 4) Primary queue is nonempty and the secondary user

misses primary activity: This case happens with a probability

PmdPr{Qp -I- O}. The secondary user will now use the

beamforming vector of the form (1) and the outage probability

becomes

(9)

Based on the preceding enumeration, the secondary mean

service rate, J.Ls, is given by:

J.Ls = (Pout,s (1 - Pfa) + P��t,s Pfa) Pr{ Qp = O}

+ (P��L(l - Pmd) + P�uts Pmd) Pr{ Qp -I- O} (10)

Our main objective in this paper is to characterize the

stability region defined as the set of arrival rate pairs (Ap, As) such that the system queues are stable. As is evident from (5)

and (10), Qp and Qs are interacting and their direct analysis

is intractable. Hence, we resort to the concept of dominant

systems as explained in the next section.

639

IV. STABILITY ANALYSIS USING DOMINANT SYSTEMS

An important performance measure of a communication

network is the stability of the queues. Stability can be defined

rigorously as follows. Denote by Q(t) the length of queue Q at the beginning of time slot t. Queue Q is said to be stable

if [7], [8] lim lim Pr{ Q(t) < x} = 1

(11) x--+oo t--+oo

In a multiqueue system, the system is stable when all queues

are stable. We can apply Loynes' theorem to check the stability

of a queue [9]. This theorem states that if the arrival process

and the service process of a queue are strictly stationary, and

the mean service rate is greater than the mean arrival rate of

the queue, then the queue is stable, otherwise it is unstable.

In order to analyze the interacting queues, we employ the

concept of dominant systems introduced in [10]. In a dominant

system a user transmits dummy packets if its queue is empty.

Since we have two users, we can construct two dominant

systems, one with the primary transmitter sending dummy

packet when Qp is empty and the other with the secondary

transmitter sending dummy packets when Qs is empty. We

explain below the relation between the stability region of both

the original and dominant systems.

A. First dominant system In the first dominant system, the primary transmitter sends

dununy packets when its queue is empty, whereas the sec­

ondary transmitter behaves as it would in the original system.

This effectively means that Pr{ Qp = O} = O. By plugging this

probability into (10), the mean service rate for the secondary

user in this system becomes

pd _ ( (d) (1 ) + md ) f..Ls - Pout,s - Pmd Pout,s Pmd (12)

where the superscript 'pd' indicates that this secondary mean

service rate corresponds to the dominant system where the

primary queue is made to transmit dummy packets when Qp is empty. Note that in the first dominant system, the secondary

mean service rate no longer depends on the state of occupancy

of Qp. Provided that the secondary queue is stable, i.e., As < f..L�d, the probability of the secondary queue being empty is

given by:

Pr{Qs = O} = 1 - AS

d f..L� (13)

Now, by substituting with (13) and (12) in (5), we obtain the

mean service rate of the primary queue as

pd _ (1 ) As ( md ) f..Lp - - Pout,p - ---p;:rPmd Pout,p - Pout,p f..Ls (14)

Based on the construction of the first dominant system it can

be noted that the queues of the dominant system are never less

in length than those of the original system, provided that they

are both initialized identically. This is because the primary

transmitter sends dummy packets even if it does not have any

packets of its own, and therefore the transmission opportunities

for the secondary transmitter are reduced. The secondary mean

service rate is thus reduced in the dominant system and Qs

is emptied less frequently. This in turn lowers the occurrence

of the event where Qs is empty and the primary transmitter

operates freely without any interference. This reduces the

primary mean service rate. Given this, if the queues are stable

in the dominant system then they are stable in the original

system. That is, the stability conditions of the dominant system

are sufficient for the stability of the original system. Now if

Qp saturates in the dominant system, the primary transmitter

will not transmit dummy packets as it always has its own

packets to send. For As < f..L�d, this makes the behavior of the

dominant system identical to that of the original system and

both systems are indistinguishable at the boundary points.

The stability conditions of the dominant system are thus both

sufficient and necessary for the stability of the original system

given that As < f..L�d. The stability region based on the first dominant system

is given by the closure of the rate pairs Ap, As constrained

by stability of the queues. One method to characterize this

closure is to solve a constrained optimization problem to find

the maximum feasible Ap corresponding to each feasible As [8], [11]. For a fixed As, the maximum stable arrival rate of

the primary transmitter is given by the following optimization

problem:

We comment now on how the transmitted power of the

secondary network affects the service rate of the primary and

secondary users. From a secondary user's point of view, it

directly enhances the received SINR of the secondary user

and thus increases f..L�d. However, increasing the secondary

transmitted power can have two contradicting influences, with

relatively different impacts depending on the values of As and

f..L�d. On one hand, increasing the transmitted power of the

secondary user can degrade the service rate of the primary

user; in the cases where the primary user is transmitting and

the secondary user misdetects the presence of the primary user,

increasing the transmitted power introduces extra interference

levels at the primary receiver and thus degrades f..Lgd. On the

other hand, increasing the transmitted power can help the

secondary user to empty its queue faster and evacuate the

wireless link, thus enhancing f..Lgd.

B. Second dominant system In the second dominant system, the secondary transmitter

sends dummy packet when its queue is empty, whereas the

primary transmitter behaves as it would in the original system.

This means that Pr{ Qs = O} = 0 from the primary user's

point of view. Using this value in (5) gives the mean service

rate for the primary user as

where superscript 'sd' refers to the dominant system where the

secondary user is made to transmit dummy packets when Qs is

640

1 st Dominant System 1 st Dominant System 1 sl Dominant Syslem

O.37�--�--��-�--�

0.9

0.8 .' ************

•• * 0.366 .*

• * .' * *.* *.* ,. t****** *.* lit. '" Hii liE* •

*.**'

: I *

',

_

0

5 1 ::r----------t/ 0.7

pdo.6

!l S 0.5

0.4

. • . .

------ , -------------------0.3 *1

0,364

!l pd P 0,362

0.36

* A _

0.1

* As .0,35

p 1.2 s,

0.8 0.6

• I 0.358 0.2

* • : P s range for Q s stability (\ = 0.35) * * *****f*7* -*-***** * ***--** I 0.4

• * I

0.' * . 0,356 • 0.2

I

0.5 p' s 1.5

0.3540:-------7:0.5=----,.-, ------:C1.5=---------! o L--�--�--�--��� o 0.2 0.4 A 0.6 O.S

Ps S

Fig. 2. Mean service rate for the secondary user versus secondary transmit power in the first dominant system.

Fig. 3. Mean service rate for the primary user ver­sus secondary transmit power in the first dominant system.

Fig. 4. Optimal secondary transmit power versus As for the first dominant system of Figures 2 and 3.

empty. Given that the primary queue is stable, i.e., Ap < fL�d, the probability of the primary queue being empty is given by:

Pr{Qp = O} = 1- Apd . (17) fL�

The secondary mean service rate can thus be written as

follows.

sd Ap ( (d) (1 ) md (1 ) fLs = fL�d Pout,s - Pmd + Pout,s Pmd - Pout,s - Pfa

- P��t,s Pfa) + (Pout,s (1 - Pfa) + P��t,s Pfa) (\8)

Following the argument provided in the previous subsection,

the second dominant system and the original system are

indistinguishable at the boundary points given that Ap < fL�d. The stability region can be obtained by solving the following

optimization problem for each Ap < fL�d: max. As = fLfd s.t. Ap < fL�d, Ps :s; Pmax (\9)

Ps

The stability region of the original system is the union of that

of both dominant systems.

V. NUMERICAL SIM UL ATION

We consider a primary user with a maximum transmission

power of Pp = 1 and the detection threshold for the primary

receiver is f3p = 1. The secondary transmitter is aided by K =

4 relays. It has a probability of misdetection and false alarm

equal to Pmd = 0.1 and Pfa = 0.01, respectively. The threshold

for detection at the secondary receiver is f3s = 1. We obtain

the outage probabilities defined in (4), (6), (7), (8) and (9)

numerically by averaging over 500,000 channel realizations.

Due to the non-convexity of the optimization problems in (15)

and (\9), we solve the problem via numerical search over the

optimal value of PSI In order to solve (\5) for a certain As, the constraint on

the stability of the secondary queue poses a lower limit on

Ps, which has an upper limit of Pmax. Figure 2 shows fLrd versus Ps in the first dominant system. Power Ps ?: 0.5 to

satisfy the condition on the stability of Qs when As = 0.35.

Generally speaking, the constraints of (\5) define the range of

secondary power for which the problem is feasible. We search

for the maximum of fL�d over this range. Figure 3 shows the

variation of fL�d with Ps at three different values for As. The

figure shows the range of Ps for which the problem is solved.

Note that the range is reduced with the lower limit on Ps increasing due to the demand for a higher power to satisfy the

stability of Qs. The optimal power is provided in Figure 4.

Power Ps is zero when no value less than or equal to Pmax can satisfy the stability condition of Qs.

We discuss now the solution of (\9). The constraint on the

stability of Qp imposes a limit on the secondary power to be

used. Since any increase in Ps produces more interference on

the primary user in the case of misdetection, then the primary

mean service rate is reduced. To satisfy the stability constraint,

Ps should be less than some value, which depends on Ap . This

is demonstrated in Figure 5 where Ps should not exceed about

0.85 for Ap = 0.35. The secondary mean service rate increases

with Ps as shown in Figure 6. Nevertheless, the range over

which we search for the optimal power value is reduced in

order to satisfy Qp stability. Figure 7 shows the optimal Ps for each possible Ap. Note that the region where Ps = 0 marks

the values of Ap at which the primary queue cannot be made

stable.

Figure 8 shows the stability regions of both dominant

systems together with their union which is the stability region

of the original system under investigation. Figure 9 compares

the stable regions achieved by two systems with Pp = 1 and Pp = 10. As expected, increasing Pp enhances the

primary arrival rate that can be operated at while preserving

the stability of the system queues. This comes at the expense

of As. VI. CONCL USION

A CRN is considered which consists of a single primary and

a single secondary links. The secondary transmitter utilizes a

set of dedicated relays by applying beamforming techniques to

null out secondary transmission at the primary receiver to al­

low for concurrent transmission with the primary user. Primary

and secondary transmitters are assumed to be equipped with

641

2nd Dominant System O,37�--�---�-'-----�--�

. 0.365 ...

0,36 sd �p 0.355

\. ---"'---------------------

..... ... ,

..... 0,35 -----..;---- - ��; - - ---.- ------

P range for Q I .... 0.345 st�bility (?. = 0.'35) : " .... ..,. .....

0.9

0.8

0.7

sd 0.6

)l S 0.5

0.4

0.3

0.2

0.'

• *

* *

: /

;0 *

2nd Dominant System

. * .

2nd Dominant System

1.S

P s

0.5

c: P )I: • 0.340:--- ----0:':.5=-------'---:-' ----:"1.5=-------:

o�. ��·-' -�--��---�--� o �--�--�--�---�-� o Ps

o 0.5 1.5 0.2 0.4 Ie p 0.6 0.8 p s

Fig. 5. Mean service rate for the primary user versus secondary transmit power in the second dominant system.

Fig. 6. Mean service rate for the secondary user versus secondary transmit power in the second dominant system.

Fig. 7. Optimal secondary transmit power versus Ap for the second dominant systems of Figures 5 and 6.

"-P 0.5

ID S�ability �egion �f the system with two Interacting queues

r--------_____ '\ O�------------------� o 0.2 0.4 "- 0.6 0.8

s

Fig. 8. Stable region of the two interacting queues system.

finite buffers, and we studied the stable region of both queues.

Sensing errors are taken into account, and their effect is shown

on the achievable stable region. We resorted to the concept

of "dominant systems" in order to decouple the interacting

queues and arrive at the provided results, and we proved

that this approach provides the exact maximum stable region.

Numerical evaluation are provided to give useful insights on

the obtained results.

VII. ACKNOWL EDGEMENT

This work has been supported in part by a grant from the

Egyptian National Telecommunication Regulatory Authority

(NTRA).

REFERENCES

[1] O. Simeone, Y Bar-Ness, and U. Spagnolini, "Stable throughput of cognitive radios with and without relaying capability," Communications, IEEE Transactions on, vol. 55, no. 12, pp. 2351-2360, 2007.

[2] J. Gambini, O. Simeone, U. Spagnolini, Y Bar-Ness, and Y Kim, "Stability analysis of a cognitive multiple access channel with primary qos constraints," in Signals, Systems and Computers, 2007. ACSSC

2007. Conference Record of the Forty-First Asilomar Conference on. IEEE, 2007, pp. 787-791.

[3] G. Zheng, S. Ma, K.K. Wong, and T.S. Ng, "Robust beamforming in cognitive radio," Wireless Communications. IEEE Transactions on, vol. 9, no. 2, pp. 570-576, 2010.

"-p

• Stability region of the system with two interacting queues (P p =

D Stability region of the system with two interacting queues (P = 1)

0.2 0.1

0 0�------0.-2------ 0- .4--- "_----0-.6------- 0 -.8----� S

Fig. 9. Stable region of the two interacting queues system for Pp = I, 10.

[4] M.A. Beigi and S.M. Razavizadeh, "Cooperative beamforming in cognitive radio networks," in Wireless Days (WD), 2009 2nd IFfP. IEEE, 2009, pp. 1-5.

[5] J. Liu, W. Chen, Z. Cao, and YJ. Zhang, "Delay optimal scheduling for cognitive radios with cooperative beamforming: A structured matrix­geometric method," Mobile Computing, IEEE Transactions on, vol. II, no. 8, pp. 1412-1423,2012.

[6] J. Liu, W. Chen, Z. Cao, and YJ.A. Zhang, "Cooperative beamforming for cognitive radio networks: A cross-layer design," Communications, IEEE Transactions on, vol. 60, no. 5, pp. 1420-1431,2012.

[7] W. Szpankowski, "Stability conditions for some distributed systems: buffered random access systems," Advances in Applied Probability, pp. 498-515, 1994.

[8] A.K. Sadek, KJ.R. Liu, and A. Ephremides, "Cognitive multiple access via cooperation: protocol design and performance analysis," IEEE

Transactions on Information Theory, vol. 53, no. 10, pp. 3677-3696, Oct. 2007.

[9] R.M. Loynes, "The stability of a queue with non-independent inter­arrival and service times," in Proc. Cambridge Philos. Soc. Cambridge University Press, 1962, vol. 58, pp. 497-520.

[10] R.R. Rao and A. Ephremides, "On the stability of interacting queues in a multiple-access system," IEEE Transactions on Information Theory, vol. 34, no. 5, pp. 918-930, Sep. 1988.

[11] S. Kompella, G.D. Nguyen, J.E. Wieselthier, and A. Ephremides, "Stable throughput tradeoffs in cognitive shared channels with cooperative relaying," in Proceedings IEEE INF O COM, Apr. 2011, pp. 1961-1969.

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