Abstract

This paper investigates a novel method for boosting the performance of parallel-relaying decode-and-forward (DF) cooperative free space optical (FSO) networks. The proposed solution takes advantage from the presence of the radio frequency (RF) links that are often established to backup the FSO links in practical systems. In this context, the low speed RF links are used for carrying the information packets that have not been delivered along the direct FSO link for the sake of sharing these packets among the relays in a simple and efficient manner. This packet sharing enhances the chances for correct detection at the relays thus increasing the number of relays that will participate in the relay-destination transmission phase. An exact outage probability analysis is carried out in the case of gamma-gamma FSO fading channels with pointing errors and of Rician/Rayleigh RF fading channels. At a second time, the cut-set method is applied for deriving a simple and tractable upper-bound that is useful for evaluating the asymptotic performance and diversity gain.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Switching-Based Cooperative Decode-and-Forward Relaying for Hybrid FSO/RF Networks

Shubha Sharma, A. S. Madhukumar, and Swaminathan R.
J. Opt. Commun. Netw. 11(6) 267-281 (2019)

Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications

Rubén Boluda-Ruiz, Antonio García-Zambrana, Beatriz Castillo-Vázquez, and Carmen Castillo-Vázquez
Opt. Express 23(3) 2600-2617 (2015)

Protocol Design and Performance Analysis of Multiuser Mixed RF and Hybrid FSO/RF Relaying With Buffers

Yasser F. Al-Eryani, Anas M. Salhab, Salam A. Zummo, and Mohamed-Slim Alouini
J. Opt. Commun. Netw. 10(4) 309-321 (2018)

References

  • View by:
  • |
  • |
  • |

  1. S. Bloom and W. S. Hartley, “The last-mile solution: hybrid FSO radio,” Whitepaper, AirFiber Inc, (2002).
  2. A. Akbulut, H. G. Ilk, and F. Ari, “Design, availability and reliability analysis on an experimental outdoor FSO/RF communication system,” in Proc. 7th IEEE Int. Conf. on Transparent Optical Networks (ICTON), 403–406 (2005).
  3. M. Usman, H.-C. Yang, and M.-S. Alouini, “Practical switching-based hybrid FSO/RF transmission and its performance analysis,” IEEE Photonics J. 6(5), 1–13 (2014).
    [Crossref]
  4. W. Zhang, S. Hranilovic, and C. Shi, “Soft-switching hybrid FSO/RF links using short-length raptor codes: design and implementation,” IEEE J. Sel. Areas Commun. 27(9), 1698–1708 (2009).
    [Crossref]
  5. A. AbdulHussein, A. Oka, T. T. Nguyen, and L. Lampe, “Rateless coding for hybrid free-space optical and radio-frequency communication,” IEEE Trans. on Wireless Commun. 9(3), 907–913 (2010).
    [Crossref]
  6. E. Zedini, H. Soury, and M.-S. Alouini, “On the performance analysis of dual-hop mixed FSO/RF systems,” IEEE Trans. on Wireless Commun. 15(5), 3679–3689 (2016).
    [Crossref]
  7. M. I. Petkovic, A. M. Cvetkovic, G. T. Djordjevic, and G. K. Karagiannidis, “Partial relay selection with outdated channel state estimation in mixed RF/FSO systems,” J. Lightwave Technol. 33(13), 2860–2867 (2015).
    [Crossref]
  8. V. Jamali, D. S. Michalopoulos, M. Uysal, and R. Schober, “Link allocation for multiuser systems with hybrid RF/FSO backhaul: Delay-limited and delay-tolerant designs,” IEEE Trans. on Wireless Commun. 15(5), 3281–3295 (2016).
    [Crossref]
  9. M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. on Wireless Commun. 7(12), 5441–5449 (2008).
    [Crossref]
  10. C. Abou-Rjeily, “Performance analysis of FSO communications with diversity methods: Add more relays or more apertures?” IEEE J. Sel. Areas Commun. 33(9), 1890–1902 (2015).
    [Crossref]
  11. C. Abou-Rjeily and Z. Noun, “Impact of inter-relay cooperation on the performance of FSO systems with any number of relays,” IEEE Trans. on Wireless Commun. 15(6), 3796–3809 (2016).
    [Crossref]
  12. H. Sandalidis, T. Tsiftsis, and G. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27(20), 4440–4445 (2009).
    [Crossref]
  13. W. Limpakom, Y.-D. Yao, and H. Man, “Outage probability analysis of wireless relay and cooperative networks in Rician fading channels with different K-factors,” in Proc. 69th IEEE Vehicular Tech. Conf. (VTC-Spring), 1–5 (2009).
  14. R. Billinton and R. N. Allan, Reliability evaluation of power systems. Springer Science & Business Media, 2nd ed. (Springer, 2013).
  15. J. Fussell and W. Vesely, “New methodology for obtaining cut sets for fault trees,” Trans. Amer. Nucl. Soc. 15, 262–263 (1972).
  16. A. Rauzy, “Toward an efficient implementation of the MOCUS algorithm,” IEEE Trans. on Reliability 52(2), 175–180 (2003).
    [Crossref]

2016 (3)

E. Zedini, H. Soury, and M.-S. Alouini, “On the performance analysis of dual-hop mixed FSO/RF systems,” IEEE Trans. on Wireless Commun. 15(5), 3679–3689 (2016).
[Crossref]

V. Jamali, D. S. Michalopoulos, M. Uysal, and R. Schober, “Link allocation for multiuser systems with hybrid RF/FSO backhaul: Delay-limited and delay-tolerant designs,” IEEE Trans. on Wireless Commun. 15(5), 3281–3295 (2016).
[Crossref]

C. Abou-Rjeily and Z. Noun, “Impact of inter-relay cooperation on the performance of FSO systems with any number of relays,” IEEE Trans. on Wireless Commun. 15(6), 3796–3809 (2016).
[Crossref]

2015 (2)

C. Abou-Rjeily, “Performance analysis of FSO communications with diversity methods: Add more relays or more apertures?” IEEE J. Sel. Areas Commun. 33(9), 1890–1902 (2015).
[Crossref]

M. I. Petkovic, A. M. Cvetkovic, G. T. Djordjevic, and G. K. Karagiannidis, “Partial relay selection with outdated channel state estimation in mixed RF/FSO systems,” J. Lightwave Technol. 33(13), 2860–2867 (2015).
[Crossref]

2014 (1)

M. Usman, H.-C. Yang, and M.-S. Alouini, “Practical switching-based hybrid FSO/RF transmission and its performance analysis,” IEEE Photonics J. 6(5), 1–13 (2014).
[Crossref]

2010 (1)

A. AbdulHussein, A. Oka, T. T. Nguyen, and L. Lampe, “Rateless coding for hybrid free-space optical and radio-frequency communication,” IEEE Trans. on Wireless Commun. 9(3), 907–913 (2010).
[Crossref]

2009 (2)

W. Zhang, S. Hranilovic, and C. Shi, “Soft-switching hybrid FSO/RF links using short-length raptor codes: design and implementation,” IEEE J. Sel. Areas Commun. 27(9), 1698–1708 (2009).
[Crossref]

H. Sandalidis, T. Tsiftsis, and G. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27(20), 4440–4445 (2009).
[Crossref]

2008 (1)

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. on Wireless Commun. 7(12), 5441–5449 (2008).
[Crossref]

2003 (1)

A. Rauzy, “Toward an efficient implementation of the MOCUS algorithm,” IEEE Trans. on Reliability 52(2), 175–180 (2003).
[Crossref]

1972 (1)

J. Fussell and W. Vesely, “New methodology for obtaining cut sets for fault trees,” Trans. Amer. Nucl. Soc. 15, 262–263 (1972).

AbdulHussein, A.

A. AbdulHussein, A. Oka, T. T. Nguyen, and L. Lampe, “Rateless coding for hybrid free-space optical and radio-frequency communication,” IEEE Trans. on Wireless Commun. 9(3), 907–913 (2010).
[Crossref]

Abou-Rjeily, C.

C. Abou-Rjeily and Z. Noun, “Impact of inter-relay cooperation on the performance of FSO systems with any number of relays,” IEEE Trans. on Wireless Commun. 15(6), 3796–3809 (2016).
[Crossref]

C. Abou-Rjeily, “Performance analysis of FSO communications with diversity methods: Add more relays or more apertures?” IEEE J. Sel. Areas Commun. 33(9), 1890–1902 (2015).
[Crossref]

Akbulut, A.

A. Akbulut, H. G. Ilk, and F. Ari, “Design, availability and reliability analysis on an experimental outdoor FSO/RF communication system,” in Proc. 7th IEEE Int. Conf. on Transparent Optical Networks (ICTON), 403–406 (2005).

Allan, R. N.

R. Billinton and R. N. Allan, Reliability evaluation of power systems. Springer Science & Business Media, 2nd ed. (Springer, 2013).

Alouini, M.-S.

E. Zedini, H. Soury, and M.-S. Alouini, “On the performance analysis of dual-hop mixed FSO/RF systems,” IEEE Trans. on Wireless Commun. 15(5), 3679–3689 (2016).
[Crossref]

M. Usman, H.-C. Yang, and M.-S. Alouini, “Practical switching-based hybrid FSO/RF transmission and its performance analysis,” IEEE Photonics J. 6(5), 1–13 (2014).
[Crossref]

Ari, F.

A. Akbulut, H. G. Ilk, and F. Ari, “Design, availability and reliability analysis on an experimental outdoor FSO/RF communication system,” in Proc. 7th IEEE Int. Conf. on Transparent Optical Networks (ICTON), 403–406 (2005).

Billinton, R.

R. Billinton and R. N. Allan, Reliability evaluation of power systems. Springer Science & Business Media, 2nd ed. (Springer, 2013).

Bloom, S.

S. Bloom and W. S. Hartley, “The last-mile solution: hybrid FSO radio,” Whitepaper, AirFiber Inc, (2002).

Cvetkovic, A. M.

Djordjevic, G. T.

Fussell, J.

J. Fussell and W. Vesely, “New methodology for obtaining cut sets for fault trees,” Trans. Amer. Nucl. Soc. 15, 262–263 (1972).

Hartley, W. S.

S. Bloom and W. S. Hartley, “The last-mile solution: hybrid FSO radio,” Whitepaper, AirFiber Inc, (2002).

Hranilovic, S.

W. Zhang, S. Hranilovic, and C. Shi, “Soft-switching hybrid FSO/RF links using short-length raptor codes: design and implementation,” IEEE J. Sel. Areas Commun. 27(9), 1698–1708 (2009).
[Crossref]

Ilk, H. G.

A. Akbulut, H. G. Ilk, and F. Ari, “Design, availability and reliability analysis on an experimental outdoor FSO/RF communication system,” in Proc. 7th IEEE Int. Conf. on Transparent Optical Networks (ICTON), 403–406 (2005).

Jamali, V.

V. Jamali, D. S. Michalopoulos, M. Uysal, and R. Schober, “Link allocation for multiuser systems with hybrid RF/FSO backhaul: Delay-limited and delay-tolerant designs,” IEEE Trans. on Wireless Commun. 15(5), 3281–3295 (2016).
[Crossref]

Karagiannidis, G.

Karagiannidis, G. K.

Lampe, L.

A. AbdulHussein, A. Oka, T. T. Nguyen, and L. Lampe, “Rateless coding for hybrid free-space optical and radio-frequency communication,” IEEE Trans. on Wireless Commun. 9(3), 907–913 (2010).
[Crossref]

Limpakom, W.

W. Limpakom, Y.-D. Yao, and H. Man, “Outage probability analysis of wireless relay and cooperative networks in Rician fading channels with different K-factors,” in Proc. 69th IEEE Vehicular Tech. Conf. (VTC-Spring), 1–5 (2009).

Man, H.

W. Limpakom, Y.-D. Yao, and H. Man, “Outage probability analysis of wireless relay and cooperative networks in Rician fading channels with different K-factors,” in Proc. 69th IEEE Vehicular Tech. Conf. (VTC-Spring), 1–5 (2009).

Michalopoulos, D. S.

V. Jamali, D. S. Michalopoulos, M. Uysal, and R. Schober, “Link allocation for multiuser systems with hybrid RF/FSO backhaul: Delay-limited and delay-tolerant designs,” IEEE Trans. on Wireless Commun. 15(5), 3281–3295 (2016).
[Crossref]

Nguyen, T. T.

A. AbdulHussein, A. Oka, T. T. Nguyen, and L. Lampe, “Rateless coding for hybrid free-space optical and radio-frequency communication,” IEEE Trans. on Wireless Commun. 9(3), 907–913 (2010).
[Crossref]

Noun, Z.

C. Abou-Rjeily and Z. Noun, “Impact of inter-relay cooperation on the performance of FSO systems with any number of relays,” IEEE Trans. on Wireless Commun. 15(6), 3796–3809 (2016).
[Crossref]

Oka, A.

A. AbdulHussein, A. Oka, T. T. Nguyen, and L. Lampe, “Rateless coding for hybrid free-space optical and radio-frequency communication,” IEEE Trans. on Wireless Commun. 9(3), 907–913 (2010).
[Crossref]

Petkovic, M. I.

Rauzy, A.

A. Rauzy, “Toward an efficient implementation of the MOCUS algorithm,” IEEE Trans. on Reliability 52(2), 175–180 (2003).
[Crossref]

Safari, M.

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. on Wireless Commun. 7(12), 5441–5449 (2008).
[Crossref]

Sandalidis, H.

Schober, R.

V. Jamali, D. S. Michalopoulos, M. Uysal, and R. Schober, “Link allocation for multiuser systems with hybrid RF/FSO backhaul: Delay-limited and delay-tolerant designs,” IEEE Trans. on Wireless Commun. 15(5), 3281–3295 (2016).
[Crossref]

Shi, C.

W. Zhang, S. Hranilovic, and C. Shi, “Soft-switching hybrid FSO/RF links using short-length raptor codes: design and implementation,” IEEE J. Sel. Areas Commun. 27(9), 1698–1708 (2009).
[Crossref]

Soury, H.

E. Zedini, H. Soury, and M.-S. Alouini, “On the performance analysis of dual-hop mixed FSO/RF systems,” IEEE Trans. on Wireless Commun. 15(5), 3679–3689 (2016).
[Crossref]

Tsiftsis, T.

Usman, M.

M. Usman, H.-C. Yang, and M.-S. Alouini, “Practical switching-based hybrid FSO/RF transmission and its performance analysis,” IEEE Photonics J. 6(5), 1–13 (2014).
[Crossref]

Uysal, M.

V. Jamali, D. S. Michalopoulos, M. Uysal, and R. Schober, “Link allocation for multiuser systems with hybrid RF/FSO backhaul: Delay-limited and delay-tolerant designs,” IEEE Trans. on Wireless Commun. 15(5), 3281–3295 (2016).
[Crossref]

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. on Wireless Commun. 7(12), 5441–5449 (2008).
[Crossref]

Vesely, W.

J. Fussell and W. Vesely, “New methodology for obtaining cut sets for fault trees,” Trans. Amer. Nucl. Soc. 15, 262–263 (1972).

Yang, H.-C.

M. Usman, H.-C. Yang, and M.-S. Alouini, “Practical switching-based hybrid FSO/RF transmission and its performance analysis,” IEEE Photonics J. 6(5), 1–13 (2014).
[Crossref]

Yao, Y.-D.

W. Limpakom, Y.-D. Yao, and H. Man, “Outage probability analysis of wireless relay and cooperative networks in Rician fading channels with different K-factors,” in Proc. 69th IEEE Vehicular Tech. Conf. (VTC-Spring), 1–5 (2009).

Zedini, E.

E. Zedini, H. Soury, and M.-S. Alouini, “On the performance analysis of dual-hop mixed FSO/RF systems,” IEEE Trans. on Wireless Commun. 15(5), 3679–3689 (2016).
[Crossref]

Zhang, W.

W. Zhang, S. Hranilovic, and C. Shi, “Soft-switching hybrid FSO/RF links using short-length raptor codes: design and implementation,” IEEE J. Sel. Areas Commun. 27(9), 1698–1708 (2009).
[Crossref]

IEEE J. Sel. Areas Commun. (2)

W. Zhang, S. Hranilovic, and C. Shi, “Soft-switching hybrid FSO/RF links using short-length raptor codes: design and implementation,” IEEE J. Sel. Areas Commun. 27(9), 1698–1708 (2009).
[Crossref]

C. Abou-Rjeily, “Performance analysis of FSO communications with diversity methods: Add more relays or more apertures?” IEEE J. Sel. Areas Commun. 33(9), 1890–1902 (2015).
[Crossref]

IEEE Photonics J. (1)

M. Usman, H.-C. Yang, and M.-S. Alouini, “Practical switching-based hybrid FSO/RF transmission and its performance analysis,” IEEE Photonics J. 6(5), 1–13 (2014).
[Crossref]

IEEE Trans. on Reliability (1)

A. Rauzy, “Toward an efficient implementation of the MOCUS algorithm,” IEEE Trans. on Reliability 52(2), 175–180 (2003).
[Crossref]

IEEE Trans. on Wireless Commun. (5)

C. Abou-Rjeily and Z. Noun, “Impact of inter-relay cooperation on the performance of FSO systems with any number of relays,” IEEE Trans. on Wireless Commun. 15(6), 3796–3809 (2016).
[Crossref]

A. AbdulHussein, A. Oka, T. T. Nguyen, and L. Lampe, “Rateless coding for hybrid free-space optical and radio-frequency communication,” IEEE Trans. on Wireless Commun. 9(3), 907–913 (2010).
[Crossref]

E. Zedini, H. Soury, and M.-S. Alouini, “On the performance analysis of dual-hop mixed FSO/RF systems,” IEEE Trans. on Wireless Commun. 15(5), 3679–3689 (2016).
[Crossref]

V. Jamali, D. S. Michalopoulos, M. Uysal, and R. Schober, “Link allocation for multiuser systems with hybrid RF/FSO backhaul: Delay-limited and delay-tolerant designs,” IEEE Trans. on Wireless Commun. 15(5), 3281–3295 (2016).
[Crossref]

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. on Wireless Commun. 7(12), 5441–5449 (2008).
[Crossref]

J. Lightwave Technol. (2)

Trans. Amer. Nucl. Soc. (1)

J. Fussell and W. Vesely, “New methodology for obtaining cut sets for fault trees,” Trans. Amer. Nucl. Soc. 15, 262–263 (1972).

Other (4)

S. Bloom and W. S. Hartley, “The last-mile solution: hybrid FSO radio,” Whitepaper, AirFiber Inc, (2002).

A. Akbulut, H. G. Ilk, and F. Ari, “Design, availability and reliability analysis on an experimental outdoor FSO/RF communication system,” in Proc. 7th IEEE Int. Conf. on Transparent Optical Networks (ICTON), 403–406 (2005).

W. Limpakom, Y.-D. Yao, and H. Man, “Outage probability analysis of wireless relay and cooperative networks in Rician fading channels with different K-factors,” in Proc. 69th IEEE Vehicular Tech. Conf. (VTC-Spring), 1–5 (2009).

R. Billinton and R. N. Allan, Reliability evaluation of power systems. Springer Science & Business Media, 2nd ed. (Springer, 2013).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1 A cooperative FSO network with three RF inter-connected relays.
Fig. 2
Fig. 2 Performance of the proposed scheme with 2, 4 and 6 relays. Solid lines correspond to the exact outage probability while the associated dotted lines correspond to the upper-bound. The same markers are used for both the exact outage probability and its corresponding upper-bound.
Fig. 3
Fig. 3 Performance of the proposed scheme with 3 and 5 relays. Solid lines correspond to the exact outage probability while the associated dotted lines correspond to the upper-bound. The same markers are used for both the exact outage probability and its corresponding upper-bound.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

f I i , j ( I ) = α i , j β i , j ξ i , j 2 A i , j I i , j ( l ) Γ ( α i , j ) Γ ( β i , j ) G 1 , 3 3 , 0 [ α i , j β i , j A i , j I i , j ( l ) I | ξ i , j 2 ξ i , j 2 1 , α i , j 1 , β i , j 1 ]
α i , j = [ exp ( 0.49 σ R , i , j 2 / ( 1 + 1.11 σ R , i , j 12 / 5 ) 7 / 6 ) 1 ] 1
β i , j = [ exp ( 0.51 σ R , i , j 2 / ( 1 + 0.69 σ R , i , j 12 / 5 ) 5 / 6 ) 1 ] 1
γ i , j = γ ¯ 0 , N + 1 ( ξ 0 , N + 1 2 + 1 A 0 , N + 1 I 0 , N + 1 ( l ) ξ 0 , N + 1 2 ) 2 ( I i , j N link ) 2
p i , j = F I i , j ( N link P M A 0 , N + 1 I 0 , N + 1 ( l ) ξ 0 , N + 1 2 ξ 0 , N + 1 2 + 1 )
F I i , j ( I ) = ξ i , j 2 Γ ( α i , j ) Γ ( β i , j ) G 2 , 4 3 , 1 [ α i , j β i , j A i , j I i , j ( i ) I | 1 , ξ i , j 2 + 1 ξ i , j 2 , α i , j , β i , j , 0 ]
p i , j = 1 exp ( Ω t h Ω i , j )
p i , j = 1 Q 1 ( 2 K i , j , 2 ( K i , j + 1 ) Ω t h Ω ¯ i , j )
P Net = p 0 , N + 1 P out
( 1 P ( R i R j )   ( R j R k ) ) = ( 1 p i , j ) ( 1 p j , k )
P ( R i R j )   ( R i R k ) = p i , j p j , k
P out ( 1 ) = [ 1 ( 1 p 0 , 1 ) ( 1 p 1 , 3 ) ] [ 1 ( 1 p 0 , 2 ) ( 1 p 2 , 3 ) ]
P out ( 2 ) = 1 ( 1 p 0 , 1 p 0 , 2 ) ( 1 p 1 , 3 p 2 , 3 )
[ ( R 0 R S 1 ) ( R 0 R S | S | ) ] [ ( R S 1 R N + 1 ) ( R S | S | R N + 1 ) ]
Q S 1 [ 1 n S p 0 , n ] [ 1 n S p n , N + 1 ]
P out = p 1 , 2 p 1 , 3 p 2 , 3 P out ( 1 ) + ( 1 p 1 , 2 ) p 1 , 3 p 2 , 3 P out ( 2 ) + p 1 , 2 ( 1 p 1 , 3 ) p 2 , 3 P out ( 3 ) + p 1 , 2 p 1 , 3 ( 1 p 2 , 3 ) P out ( 4 ) + ( 1 p 1 , 2 ) ( 1 p 1 , 3 ) p 2 , 3 P out ( 5 ) + ( 1 p 1 , 2 ) p 1 , 3 ( 1 p 2 , 3 ) P out ( 6 ) + p 1 , 2 ( 1 p 1 , 3 ) ( 1 p 2 , 3 ) P out 7 + ( 1 p 1 , 2 ) ( 1 p 1 , 3 ) ( 1 p 2 , 3 ) P out ( 8 )
F [ f 1 , 2 , , f 1 , N , f 2 , 3 , , f 2 , N , , f N 1 , N ]
f n , n = { 0 , link R n - R n in outage ; 1 , link R n - R n not in outage .
P out = F { 0 , 1 } ( 2 N ) π ( F ) m = 1 M Q S m ( F )
π ( f n , n ) = { p n , n , f n , n = 0 ; 1 p n , n , f n , n = 1 .
( n , n ) S m ( F ) × S m ( F ) ( with m m ) : f n , n = 0
( n , n ) S m ( F ) × S m ( F ) ( with n n ) : K S m ( F ) | f n , K 1 = f K 1 , K 2 = = f K | K | , n = 1
P out P U . B . i = 1 m Pr ( C i in outage )
P U . B . = p 0 , 1 p 0 , 2 + p 1 , 3 p 2 , 3 + p 0 , 1 p 1 , 2 p 2 , 3 + p 0 , 2 p 1 , 2 p 1 , 3
P U . B . = n = 0 N i = 1 ( N n ) [ j n , i p j , N + 1 ] [ j ¯ n , i p 0 , j ] [ j n , i ; j ¯ n , i p j , j ]
P U . B . = P U . B . ( 0 ) + P U . B . ( 1 ) + P U . B . ( 2 ) + P U . B . ( 3 ) + P U . B . ( 4 )
P U . B . ( 1 ) = p 1 , 5 ( p 0 , 2 p 0 , 3 p 0 , 4 ) ( p 1 , 2 p 1 , 3 p 1 , 4 ) + p 2 , 5 ( p 0 , 1 p 0 , 3 p 0 , 4 ) ( p 2 , 1 p 2 , 3 p 2 , 4 ) + p 3 , 5 ( p 0 , 1 p 0 , 2 p 0 , 4 ) ( p 3 , 1 p 3 , 2 p 3 , 4 ) + p 4 , 5 ( p 0 , 1 p 0 , 2 p 0 , 3 ) ( p 4 , 1 p 4 , 2 p 4 , 3 )
P U . B . ( 2 ) = ( p 1 , 5 p 2 , 5 ) ( p 0 , 3 p 0 , 4 ) ( p 1 , 3 p 1 , 4 p 2 , 3 p 2 , 4 ) + ( p 1 , 5 p 3 , 5 ) ( p 0 , 2 p 0 , 4 ) ( p 1 , 2 p 1 , 4 p 3 , 2 p 3 , 4 ) + ( p 1 , 5 p 4 , 5 ) ( p 0 , 2 p 0 , 3 ) ( p 1 , 2 p 1 , 3 p 4 , 2 p 4 , 3 ) + ( p 2 , 5 p 3 , 5 ) ( p 0 , 1 p 0 , 4 ) ( p 2 , 1 p 2 , 4 p 3 , 1 p 3 , 4 ) + ( p 2 , 5 p 4 , 5 ) ( p 0 , 1 p 0 , 3 ) ( p 2 , 1 p 2 , 3 p 4 , 1 p 4 , 3 ) + ( p 3 , 5 p 4 , 5 ) ( p 0 , 1 p 0 , 2 ) ( p 3 , 1 p 3 , 2 p 4 , 1 p 4 , 2 )
P U . B . ( 3 ) = ( p 1 , 5 p 2 , 5 p 3 , 5 ) p 0 , 4 ( p 1 , 4 p 2 , 4 p 3 , 4 ) + ( p 1 , 5 p 2 , 5 p 4 , 5 ) p 0 , 3 ( p 1 , 3 p 2 , 3 p 4 , 3 ) + ( p 1 , 5 p 3 , 5 p 4 , 5 ) p 0 , 2 ( p 1 , 2 p 3 , 2 p 4 , 2 ) + ( p 2 , 5 p 3 , 5 p 4 , 5 ) p 0 , 1 ( p 2 , 1 p 3 , 1 p 4 , 1 )
P Net P 0 , N + 1 [ n = 1 N P 0 , n + n = 1 N P n , N + 1 ]
P Net ( 0 ) p 0 , N + 1 n = 1 N [ p 0 , n + p n , N + 1 ]
ζ Net = ζ 0 , N + 1 + min { n = 1 N ζ 0 , n , n = 1 N ζ n , N + 1 }
ζ Net ( 0 ) = ζ 0 , N + 1 + n = 1 N min { ζ 0 , n , ζ n , N + 1 }

Metrics