Queuing Theory

1. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis in a Cognitive Radio
System with Cooperative Beamforming
Reference: Karmoose, Mohammed, Ahmed Sultan, and Moustafa Youssef. "Stability analysis in a
cognitive radio system with cooperative beamforming." Wireless Communications and Networking
Conference (WCNC), 2013 IEEE. IEEE, 2013.
Mohamed Seif1
, and Abdelrahman Youssef1
1Wireless Intelligent Networks Center (WINC), Nile University, Egypt
June 22, 2015
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 1

2. Outline System Model System Analysis Numerical Results Conclusion
Outline
1 Outline
2 System Model
Network Model
Sensing and Beamforming
3 System Analysis
Queue Service Rates
Stability Analysis
4 Numerical Results
Numerical Results
5 Conclusion
Conclusion
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 2

3. Outline System Model System Analysis Numerical Results Conclusion
Outline
•System Model
•System Analysis
M. Seif
•Numerical Results
•Conclusions
A. Youssef
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 3

4. Outline System Model System Analysis Numerical Results Conclusion
Network Model
Network Model
Pair of a SU are communicating in
the presence of 2 PUs, each
equipped with one antenna
K (decode and forward) relays are
working together forming a virtual
antenna array (VAA)
Functions of relays:
1 Enhance the throughput of SU
network
2 Null the interference on PU
network
TX RX
1
2
K
TX RX
Secondary User Network
Primary User Network
Figure: CRN model
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 4

5. Outline System Model System Analysis Numerical Results Conclusion
Sensing and Beamforming
Sensing and Beamforming
SU-TX senses the occupancy of PU
activity
1 If PU is idle:
ws =
√
Ps
Hs
Hs
2 If PU is active:
ws =
√
Ps
(I−φ)Hs
√
HH
s (I−φ)Hs
TX RX
1
2
K
TX RX
Secondary User Network
Primary User Network
Figure: CRN model
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 5

6. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
A. Primary User Service Rate
1 Secondary user’s queue is empty:
pout,p = Pr{Pp Hp
2
< βp} = 1 − exp(
βp
Pp
)
Then, primary service rate is 1 − pout,p
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6

7. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
A. Primary User Service Rate
1 Secondary user’s queue is empty:
pout,p = Pr{Pp Hp
2
< βp} = 1 − exp(
βp
Pp
)
Then, primary service rate is 1 − pout,p
2 Secondary user’s is nonempty and the primary user is detected:
This event happens w.p. (1 − pmd )Pr{Qs ≠ 0}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6

8. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
A. Primary User Service Rate
1 Secondary user’s queue is empty:
pout,p = Pr{Pp Hp
2
< βp} = 1 − exp(
βp
Pp
)
Then, primary service rate is 1 − pout,p
2 Secondary user’s is nonempty and the primary user is detected:
This event happens w.p. (1 − pmd )Pr{Qs ≠ 0}
Then, primary service rate is 1 − pout,p
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6

9. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
A. Primary User Service Rate
1 Secondary user’s queue is empty:
pout,p = Pr{Pp Hp
2
< βp} = 1 − exp(
βp
Pp
)
Then, primary service rate is 1 − pout,p
2 Secondary user’s is nonempty and the primary user is detected:
This event happens w.p. (1 − pmd )Pr{Qs ≠ 0}
Then, primary service rate is 1 − pout,p
3 Secondary user’s queue is nonempty and the primary user is
misdetected:
This event happens w.p. pmd Pr{Qs ≠ 0}
Then, primary service rate is 1 − pout,p
The relays mistakenly don’t employ the nulling
beamforming vector
pmd
out,p = Pr{
Pp Hp
2
HH
spwa +1
< βp}
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6

10. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
A. Primary User Service Rate
The mean service rate for PU is:
µp = (1 − pout,p)(Pr{Qs = 0} + (1 − pmd )Pr{Qs ≠ 0}) + (1 − p
md
out,p)(pmd Pr{Qs ≠ 0})
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 7

11. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
B. Secondary User Service Rate
1 Primary Queue is empty and the secondary user detects the
channel to be vacant:
This case happens w.p. (1 − pfa)Pr{Qp = 0}
pout,s = Pr{Ps Hs
2
< βs}
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 8

12. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
B. Secondary User Service Rate
1 Primary Queue is empty and the secondary user detects the
channel to be vacant:
This case happens w.p. (1 − pfa)Pr{Qp = 0}
pout,s = Pr{Ps Hs
2
< βs}
2 Primary queue is empty and the secondary user finds the
channel busy:
This case happens w.p. pfaPr{Qp = 0}
pfa
out,s = Pr{ HH
s wp
2
< βs}
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 8

13. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
B. Secondary User Service Rate
1 Primary Queue is empty and the secondary user detects the
channel to be vacant:
This case happens w.p. (1 − pfa)Pr{Qp = 0}
pout,s = Pr{Ps Hs
2
< βs}
2 Primary queue is empty and the secondary user finds the
channel busy:
This case happens w.p. pfaPr{Qp = 0}
pfa
out,s = Pr{ HH
s wp
2
< βs}
3 Primary queue is nonempty and the secondary user detects
primary activity:
This case happens w.p. (1 − pmd )Pr{Qp ≠ 0}
pd
out,s = Pr{
HH
s wp
2
Pp Hps 2+1
< βs}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 8

14. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
B. Secondary User Service Rate
1 Primary Queue is empty and the secondary user detects the
channel to be vacant:
This case happens w.p. (1 − pfa)Pr{Qp = 0}
pout,s = Pr{Ps Hs
2
< βs}
2 Primary queue is empty and the secondary user finds the
channel busy:
This case happens w.p. pfaPr{Qp = 0}
pfa
out,s = Pr{ HH
s wp
2
< βs}
3 Primary queue is nonempty and the secondary user detects
primary activity:
This case happens w.p. (1 − pmd )Pr{Qp ≠ 0}
pd
out,s = Pr{
HH
s wp
2
Pp Hps 2+1
< βs}
4 Primary queue is nonempty and the secondary user misdetects
primary activity:
This case happens w.p. pmd Pr{Qp ≠ 0}, pd
out,s = Pr{
Ps Hs
2
Pp Hps 2+1
< βs}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 8

15. Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
B. Secondary User Service Rate
The mean service rate for SU is:
(pout,s(1 − pfa) + p
fa
out,spfa)Pr{Qp = 0} + (P
d
out,s(1 − pmd ) + p
md
out,spmd )Pr{Qp ≠ 0}
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 9

16. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis
Our main objective is to characterize
the stability region defined as the
set of arival pairs (λp,λs)
Since Qp and Qs are interacting
togheter and their direct analysis is
itractable
The concept of dominant systems
TX RX
1
2
K
TX RX
Secondary User Network
Primary User Network
Figure: CRN model
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 10

17. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis
In a multiqueue system, the system is stable when all queues
are stable. We can apply Loynes’theorem to check the stability
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 11

18. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis
In a multiqueue system, the system is stable when all queues
are stable. We can apply Loynes’theorem to check the stability
Theorem
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.
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 11

19. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis using Dominant Systems
In order to analyze the interacting queues, we employ the
concept of dominant systems.
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 12

20. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis using Dominant Systems
In order to analyze the interacting queues, we employ the
concept of dominant systems.
In a dominant system a user transmits dummy packets if its
queue is empty
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 12

21. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis using Dominant Systems
In order to analyze the interacting queues, we employ the
concept of dominant systems.
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
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 12

22. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The primary transmitter sends dummy packets when its queue is
empty
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 13

23. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The primary transmitter sends dummy packets when its queue is
empty
Whereas, the secondart transmitter behaves as it would in the
original system
Pr{Qp = 0} = 0
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 13

24. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The primary transmitter sends dummy packets when its queue is
empty
Whereas, the secondart transmitter behaves as it would in the
original system
Pr{Qp = 0} = 0
µpd
s = (pd
out,s(1 − pmd ) + pmd
out,spmd )
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 13

25. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The primary transmitter sends dummy packets when its queue is
empty
Whereas, the secondary transmitter behaves as it would in the
original system
Pr{Qs = 0} = 1 − λs
µpd
s
µpd
p = (1 − pout,p) − λs
µpd
s
pmd (pmd
out,p − pout,p)
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 14

26. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The stability region on the first dominant system is given by the
closure of the rate pairs (λp,λs)
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 15

27. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The stability region on the first dominant system is given by the
closure of the rate pairs (λp,λs)
max λp = µpd
p s.t. λs < µpd
s ,Ps ≤ Pmax
Same manner for the second dominant system
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 15

28. Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The stability region on the first dominant system is given by the
closure of the rate pairs (λp,λs)
max λp = µpd
p s.t. λs < µpd
s ,Ps ≤ Pmax
Same manner for the second dominant system
Stability region of the original system is the union of the two
dominant system (Theorem)
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 15

29. Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 16

30. Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 17

31. Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Figure: Optimal secondary transmit power versus λs for the first
dominant system
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Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 18

32. Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 19

33. Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 20

34. Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Figure: Optimal secondary transmit power versus λp for the second
dominant systems
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 21

35. Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 22

36. Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 23

37. Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Conclusion
A CRN is considered which consists of a single primary and a
single secondary links.
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 24

38. Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Conclusion
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
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 24

39. Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Conclusion
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
Studying the stability region of the queues with sensing error
taken into account
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 24

40. Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Conclusion
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
Studying the stability region of the queues with sensing error
taken into account
We resorted to the concept of dominant systems in order to
decouple the interacting queues
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 24

41. Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Thank You!
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 25