stability analysis in a cognitive radio system with cooperative beamforming
TRANSCRIPT
![Page 1: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/1.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis in a Cognitive RadioSystem with Cooperative Beamforming
Reference: Karmoose, Mohammed, Ahmed Sultan, and Moustafa Youssef. "Stability analysis in acognitive 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
![Page 2: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/2.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Outline
1 Outline
2 System ModelNetwork ModelSensing and Beamforming
3 System AnalysisQueue Service RatesStability Analysis
4 Numerical ResultsNumerical Results
5 ConclusionConclusion
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 2
![Page 3: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/3.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Outline
•System Model
•System Analysis
M. Seif
•Numerical Results
•Conclusions
A. Youssef
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 3
![Page 4: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/4.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Network Model
Network Model
Pair of a SU are communicating inthe presence of 2 PUs, eachequipped with one antenna
K (decode and forward) relays areworking together forming a virtualantenna array (VAA)
Functions of relays:
1 Enhance the throughput of SUnetwork
2 Null the interference on PUnetwork
TX RX
1
2
K
TX RX
Secondary User Network
Primary User Network
Figure: CRN model
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 4
![Page 5: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/5.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Sensing and Beamforming Scheme
Sensing and Beamforming Scheme
SU-TX senses the occupancy of PUactivity
1 If PU is idle:
wa =√
PsHs∣∣Hs ∣∣
2 If PU is active:
wp =√
Ps(I−φ)Hs
√
HHs (I−φ)Hs
TX RX
1
2
K
TX RX
Secondary User Network
Primary User Network
Figure: CRN model
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 5
![Page 6: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/6.jpg)
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 ismisdetected:
This event happens w.p. pmdPr{Qs ≠ 0}The relays mistakenly don’t employ the nullingbeamforming vectorpmd
out,p = Pr{ Pp ∣Hp ∣2
∣HHspwa∣+1 < βp}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6
![Page 7: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/7.jpg)
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 ismisdetected:
This event happens w.p. pmdPr{Qs ≠ 0}The relays mistakenly don’t employ the nullingbeamforming vectorpmd
out,p = Pr{ Pp ∣Hp ∣2
∣HHspwa∣+1 < βp}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6
![Page 8: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/8.jpg)
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 ismisdetected:
This event happens w.p. pmdPr{Qs ≠ 0}The relays mistakenly don’t employ the nullingbeamforming vectorpmd
out,p = Pr{ Pp ∣Hp ∣2
∣HHspwa∣+1 < βp}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6
![Page 9: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/9.jpg)
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 ismisdetected:
This event happens w.p. pmdPr{Qs ≠ 0}The relays mistakenly don’t employ the nullingbeamforming vectorpmd
out,p = Pr{ Pp ∣Hp ∣2
∣HHspwa∣+1 < βp}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6
![Page 10: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/10.jpg)
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 ismisdetected:
This event happens w.p. pmdPr{Qs ≠ 0}The relays mistakenly don’t employ the nullingbeamforming vectorpmd
out,p = Pr{ Pp ∣Hp ∣2
∣HHspwa∣+1 < βp}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 6
![Page 11: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/11.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
A. Primary User Service Rate
The average mean service rate for PU is:
µp = (1 − pout,p)(Pr{Qs = 0} + (1 − pmd )Pr{Qs ≠ 0}) + (1 − pmdout,p)(pmd Pr{Qs ≠ 0})
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 7
![Page 12: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/12.jpg)
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 thechannel 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 thechannel busy:
This case happens w.p. pfaPr{Qp = 0}
pfaout,s = Pr{∣HH
s wp ∣2< βs}
3 Primary queue is nonempty and the secondary user detectsprimary activity:
This case happens w.p. (1 − pmd )Pr{Qp ≠ 0}
pdout,s = Pr{
∣HHs wp ∣
2
Pp ∣Hps ∣2+1< βs}
4 Primary queue is nonempty and the secondary user misdetectsprimary activity:
This case happens w.p. pmd Pr{Qp ≠ 0}, pmdout,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
![Page 13: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/13.jpg)
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 thechannel 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 thechannel busy:
This case happens w.p. pfaPr{Qp = 0}
pfaout,s = Pr{∣HH
s wp ∣2< βs}
3 Primary queue is nonempty and the secondary user detectsprimary activity:
This case happens w.p. (1 − pmd )Pr{Qp ≠ 0}
pdout,s = Pr{
∣HHs wp ∣
2
Pp ∣Hps ∣2+1< βs}
4 Primary queue is nonempty and the secondary user misdetectsprimary activity:
This case happens w.p. pmd Pr{Qp ≠ 0}, pmdout,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
![Page 14: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/14.jpg)
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 thechannel 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 thechannel busy:
This case happens w.p. pfaPr{Qp = 0}
pfaout,s = Pr{∣HH
s wp ∣2< βs}
3 Primary queue is nonempty and the secondary user detectsprimary activity:
This case happens w.p. (1 − pmd )Pr{Qp ≠ 0}
pdout,s = Pr{
∣HHs wp ∣
2
Pp ∣Hps ∣2+1< βs}
4 Primary queue is nonempty and the secondary user misdetectsprimary activity:
This case happens w.p. pmd Pr{Qp ≠ 0}, pmdout,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
![Page 15: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/15.jpg)
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 thechannel 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 thechannel busy:
This case happens w.p. pfaPr{Qp = 0}
pfaout,s = Pr{∣HH
s wp ∣2< βs}
3 Primary queue is nonempty and the secondary user detectsprimary activity:
This case happens w.p. (1 − pmd )Pr{Qp ≠ 0}
pdout,s = Pr{
∣HHs wp ∣
2
Pp ∣Hps ∣2+1< βs}
4 Primary queue is nonempty and the secondary user misdetectsprimary activity:
This case happens w.p. pmd Pr{Qp ≠ 0}, pmdout,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
![Page 16: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/16.jpg)
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 thechannel 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 thechannel busy:
This case happens w.p. pfaPr{Qp = 0}
pfaout,s = Pr{∣HH
s wp ∣2< βs}
3 Primary queue is nonempty and the secondary user detectsprimary activity:
This case happens w.p. (1 − pmd )Pr{Qp ≠ 0}
pdout,s = Pr{
∣HHs wp ∣
2
Pp ∣Hps ∣2+1< βs}
4 Primary queue is nonempty and the secondary user misdetectsprimary activity:
This case happens w.p. pmd Pr{Qp ≠ 0}, pmdout,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
![Page 17: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/17.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Queue Service Rates
B. Secondary User Service Rate
The average mean service rate for SU is:
(pout,s(1 − pfa) + pfaout,spfa)Pr{Qp = 0} + (Pd
out,s(1 − pmd ) + pmdout,spmd )Pr{Qp ≠ 0}
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 9
![Page 18: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/18.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis
Our main objective is to characterizethe stability region defined as theset of arrival pairs (λp, λs)
Since Qp and Qs are interactingtogheter, so their direct analysis isintractable
The concept of dominant systems
TX RX
1
2
K
TX RX
Secondary User Network
Primary User Network
Figure: CRN model
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 10
![Page 19: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/19.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis
In a multiqueue system, the system is stable when all queuesare stable. We can apply Loynes’theorem to check the stability
TheoremIf the arrival process and the service process of a queue are strictlystationary, and the mean service rate is greater than the mean arrivalrate of the queue, then the queue is stable, otherwise it is unstable.
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 11
![Page 20: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/20.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis
In a multiqueue system, the system is stable when all queuesare stable. We can apply Loynes’theorem to check the stability
TheoremIf the arrival process and the service process of a queue are strictlystationary, and the mean service rate is greater than the mean arrivalrate of the queue, then the queue is stable, otherwise it is unstable.
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 11
![Page 21: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/21.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis using Dominant Systems
In order to analyze the interacting queues, we employ theconcept of dominant systems.
In a dominant system a user transmits dummy packets if itsqueue is empty
Since we have two users, we can construct two dominantsystems
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 12
![Page 22: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/22.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis using Dominant Systems
In order to analyze the interacting queues, we employ theconcept of dominant systems.
In a dominant system a user transmits dummy packets if itsqueue is empty
Since we have two users, we can construct two dominantsystems
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 12
![Page 23: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/23.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
Stability Analysis using Dominant Systems
In order to analyze the interacting queues, we employ theconcept of dominant systems.
In a dominant system a user transmits dummy packets if itsqueue is empty
Since we have two users, we can construct two dominantsystems
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 12
![Page 24: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/24.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The primary transmitter sends dummy packets when its queue isempty
Whereas, the secondary transmitter behaves as it would in theoriginal system
Pr{Qp = 0} = 0
µpds = (pd
out,s(1 − pmd) + pmdout,spmd)
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 13
![Page 25: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/25.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The primary transmitter sends dummy packets when its queue isempty
Whereas, the secondary transmitter behaves as it would in theoriginal system
Pr{Qp = 0} = 0
µpds = (pd
out,s(1 − pmd) + pmdout,spmd)
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 13
![Page 26: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/26.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The primary transmitter sends dummy packets when its queue isempty
Whereas, the secondary transmitter behaves as it would in theoriginal system
Pr{Qp = 0} = 0
µpds = (pd
out,s(1 − pmd) + pmdout,spmd)
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 13
![Page 27: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/27.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Stability Analysis
I. First dominant system
The primary transmitter sends dummy packets when its queue isempty
Whereas, the secondary transmitter behaves as it would in theoriginal system
Pr{Qs = 0} = 1 − λs
µpds
µpdp = (1 − pout,p) −
λs
µpds
pmd(pmdout,p − pout,p)
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 14
![Page 28: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/28.jpg)
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 theclosure of the rate pairs (λp, λs)
max λp = µpdp s.t. λs < µ
pds ,Ps ≤ Pmax
Same manner for the second dominant system ,
Stability region of the original system is the union of the twodominant system (Theorem)
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 15
![Page 29: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/29.jpg)
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 theclosure of the rate pairs (λp, λs)
max λp = µpdp s.t. λs < µ
pds ,Ps ≤ Pmax
Same manner for the second dominant system ,
Stability region of the original system is the union of the twodominant system (Theorem)
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 15
![Page 30: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/30.jpg)
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 theclosure of the rate pairs (λp, λs)
max λp = µpdp s.t. λs < µ
pds ,Ps ≤ Pmax
Same manner for the second dominant system ,
Stability region of the original system is the union of the twodominant system (Theorem)
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 15
![Page 31: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/31.jpg)
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
![Page 32: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/32.jpg)
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 17
![Page 33: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/33.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Figure: Optimal secondary transmit power versus λs for the firstdominant system
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 18
![Page 34: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/34.jpg)
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
![Page 35: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/35.jpg)
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
![Page 36: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/36.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Numerical Results
Simulation Results
Figure: Optimal secondary transmit power versus λp for the seconddominant systems
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 21
![Page 37: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/37.jpg)
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
![Page 38: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/38.jpg)
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
![Page 39: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/39.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Conclusion
A CRN is considered which consists of 2 PUs and 2 SUs
The secondary transmitter utilizes a set of dedicated relays byapplying beamforming techniques to null out secondarytransmission at the primary receiver
Studying the stability region of the queues with sensing errorstaken into account
We resorted to the concept of dominant systems in order todecouple the interacting queues
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 24
![Page 40: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/40.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Conclusion
A CRN is considered which consists of 2 PUs and 2 SUs
The secondary transmitter utilizes a set of dedicated relays byapplying beamforming techniques to null out secondarytransmission at the primary receiver
Studying the stability region of the queues with sensing errorstaken into account
We resorted to the concept of dominant systems in order todecouple the interacting queues
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 24
![Page 41: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/41.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Conclusion
A CRN is considered which consists of 2 PUs and 2 SUs
The secondary transmitter utilizes a set of dedicated relays byapplying beamforming techniques to null out secondarytransmission at the primary receiver
Studying the stability region of the queues with sensing errorstaken into account
We resorted to the concept of dominant systems in order todecouple the interacting queues
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 24
![Page 42: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/42.jpg)
Outline System Model System Analysis Numerical Results Conclusion
Conclusion
Conclusion
A CRN is considered which consists of 2 PUs and 2 SUs
The secondary transmitter utilizes a set of dedicated relays byapplying beamforming techniques to null out secondarytransmission at the primary receiver
Studying the stability region of the queues with sensing errorstaken into account
We resorted to the concept of dominant systems in order todecouple the interacting queues
Mohamed Seif, and Abdelrahman Youssef Nile University
Stability Analysis in a Cognitive Radio System with Cooperative Beamforming 24
![Page 43: Stability Analysis in a Cognitive Radio System with Cooperative Beamforming](https://reader034.vdocuments.net/reader034/viewer/2022042716/55c14226bb61ebbe7c8b4607/html5/thumbnails/43.jpg)
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