Abstract

A crucial issue in optical packet switched (OPS) networks is packet losses at the network layer caused by contentions. This paper presents the network layer packet redundancy scheme (NLPRS), which is a novel approach to reduce the end-to-end data packet loss rate in OPS networks. By introducing redundancy packets in the OPS network, the NLPRS enables a possible reconstruction of data packets that are lost due to contentions. An analytical model of the NLPRS based on reduced load Erlang fix-point analysis is presented. Simulations of an OPS ring network show that the NLPRS is in particular efficient in small networks operating at low system loads. Results also show how the arrival process, packet length distribution, network size and redundancy packet scheduling mechanism influence the NLPRS performance.

© 2004 Optical Society of America

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References

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  1. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, 2002).
  2. M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39(3) (2001) 128–135.
    [Crossref]
  3. L. Dittmann et al., “The European IST Project DAVID: A Viable Approach Toward Optical Packet Switching,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1026–1040.
    [Crossref]
  4. J. S. Turner, “Terabit burst switching,” Journal of High Speed Networks 8(1) (1999) 3–16.
  5. M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP-Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18(10) (2000) 2062–2071.
    [Crossref]
  6. S. Yao, B. Mukherjee, S. J. Ben Yoo, and S. Dixit, “A Unified Study of Contention-Resolution Schemes in Optical Packet-Switched Networks,” IEEE Journal of Lightwave Technology 21(3) (2003) 672–683.
    [Crossref]
  7. S. L. Danielsen, C. Joergensen, B. Mikkelsen, and K. E. Stubkjaer, “Optical Packet Switched Network Layer Without Optical Buffers,” IEEE Photonics Technology Letters 10(6) (1998) 896–898.
    [Crossref]
  8. Y. Chen, H. Wu, D. Xu, and C. Qiao, “Performance Analysis of Optical Burst Switched Node with Deflection Routing,” in Proceedings of International Conference on Communication, pp. 1355–1359, 2003.
  9. D. K. Hunter, M. C. Chia, and I. Andonovic, “Buffering in Optical Packet Switches,” IEEE Journal of Lightwave Technology 16(12) (1998) 2081–2094.
    [Crossref]
  10. A. S. Tanenbaum, Computer Networks (Prentice Hall, 1996).
  11. V. Santonja, “Dependability models of RAID using stochastic activity networks,” in Proceedings of Dependable Computing Conference, pp. 141–158, 1996.
  12. Fluid Studios, FSRaid documentation, http://www.fluidstudios.com/fsraid.html (accessed March 2004).
  13. The Smart Par Primer, http://usenethelp.codeccorner.com/SPar_Primer.html (accessed March 2004).
  14. Leonard Kleinrock, Queuing Systems Volume I: Theory (John Wiley & Sons, 1975).
  15. Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance Analyses of Optical Burst-Switching Networks,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1187–1197.
    [Crossref]
  16. G. Birtwistle, DEMOS: Discrete Event Modelling on Simula (MacMillan, 1978).
  17. E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2003).
  18. H. Øverby and N. Stol, “Effects of bursty traffic in service differentiated Optical Packet Switched networks,” Optics Express 12(3) (2004) 410–415.
    [Crossref] [PubMed]
  19. H. W. Braun, NLANR/Measurement and Network analysis http://www.caida.org/analysis/AIX/plen_hist/.

2004 (1)

H. Øverby and N. Stol, “Effects of bursty traffic in service differentiated Optical Packet Switched networks,” Optics Express 12(3) (2004) 410–415.
[Crossref] [PubMed]

2003 (3)

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance Analyses of Optical Burst-Switching Networks,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1187–1197.
[Crossref]

S. Yao, B. Mukherjee, S. J. Ben Yoo, and S. Dixit, “A Unified Study of Contention-Resolution Schemes in Optical Packet-Switched Networks,” IEEE Journal of Lightwave Technology 21(3) (2003) 672–683.
[Crossref]

L. Dittmann et al., “The European IST Project DAVID: A Viable Approach Toward Optical Packet Switching,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1026–1040.
[Crossref]

2001 (1)

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39(3) (2001) 128–135.
[Crossref]

2000 (1)

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP-Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18(10) (2000) 2062–2071.
[Crossref]

1999 (1)

J. S. Turner, “Terabit burst switching,” Journal of High Speed Networks 8(1) (1999) 3–16.

1998 (2)

S. L. Danielsen, C. Joergensen, B. Mikkelsen, and K. E. Stubkjaer, “Optical Packet Switched Network Layer Without Optical Buffers,” IEEE Photonics Technology Letters 10(6) (1998) 896–898.
[Crossref]

D. K. Hunter, M. C. Chia, and I. Andonovic, “Buffering in Optical Packet Switches,” IEEE Journal of Lightwave Technology 16(12) (1998) 2081–2094.
[Crossref]

Ackaert, A.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2003).

Andonovic, I.

D. K. Hunter, M. C. Chia, and I. Andonovic, “Buffering in Optical Packet Switches,” IEEE Journal of Lightwave Technology 16(12) (1998) 2081–2094.
[Crossref]

Birtwistle, G.

G. Birtwistle, DEMOS: Discrete Event Modelling on Simula (MacMillan, 1978).

Braun, H. W.

H. W. Braun, NLANR/Measurement and Network analysis http://www.caida.org/analysis/AIX/plen_hist/.

Breusegem, E. V.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2003).

Chen, Y.

Y. Chen, H. Wu, D. Xu, and C. Qiao, “Performance Analysis of Optical Burst Switched Node with Deflection Routing,” in Proceedings of International Conference on Communication, pp. 1355–1359, 2003.

Cheyns, J.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2003).

Chia, M. C.

D. K. Hunter, M. C. Chia, and I. Andonovic, “Buffering in Optical Packet Switches,” IEEE Journal of Lightwave Technology 16(12) (1998) 2081–2094.
[Crossref]

Danielsen, S. L.

S. L. Danielsen, C. Joergensen, B. Mikkelsen, and K. E. Stubkjaer, “Optical Packet Switched Network Layer Without Optical Buffers,” IEEE Photonics Technology Letters 10(6) (1998) 896–898.
[Crossref]

Demeester, P.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2003).

Dittmann, L.

L. Dittmann et al., “The European IST Project DAVID: A Viable Approach Toward Optical Packet Switching,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1026–1040.
[Crossref]

Dixit, S.

S. Yao, B. Mukherjee, S. J. Ben Yoo, and S. Dixit, “A Unified Study of Contention-Resolution Schemes in Optical Packet-Switched Networks,” IEEE Journal of Lightwave Technology 21(3) (2003) 672–683.
[Crossref]

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP-Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18(10) (2000) 2062–2071.
[Crossref]

Hunter, D. K.

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39(3) (2001) 128–135.
[Crossref]

D. K. Hunter, M. C. Chia, and I. Andonovic, “Buffering in Optical Packet Switches,” IEEE Journal of Lightwave Technology 16(12) (1998) 2081–2094.
[Crossref]

Joergensen, C.

S. L. Danielsen, C. Joergensen, B. Mikkelsen, and K. E. Stubkjaer, “Optical Packet Switched Network Layer Without Optical Buffers,” IEEE Photonics Technology Letters 10(6) (1998) 896–898.
[Crossref]

Kleinrock, Leonard

Leonard Kleinrock, Queuing Systems Volume I: Theory (John Wiley & Sons, 1975).

Lannoo, B.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2003).

Mikkelsen, B.

S. L. Danielsen, C. Joergensen, B. Mikkelsen, and K. E. Stubkjaer, “Optical Packet Switched Network Layer Without Optical Buffers,” IEEE Photonics Technology Letters 10(6) (1998) 896–898.
[Crossref]

Mukherjee, B.

S. Yao, B. Mukherjee, S. J. Ben Yoo, and S. Dixit, “A Unified Study of Contention-Resolution Schemes in Optical Packet-Switched Networks,” IEEE Journal of Lightwave Technology 21(3) (2003) 672–683.
[Crossref]

O’Mahony, M. J.

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39(3) (2001) 128–135.
[Crossref]

Øverby, H.

H. Øverby and N. Stol, “Effects of bursty traffic in service differentiated Optical Packet Switched networks,” Optics Express 12(3) (2004) 410–415.
[Crossref] [PubMed]

Pickavet, M.

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2003).

Qiao, C.

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP-Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18(10) (2000) 2062–2071.
[Crossref]

Y. Chen, H. Wu, D. Xu, and C. Qiao, “Performance Analysis of Optical Burst Switched Node with Deflection Routing,” in Proceedings of International Conference on Communication, pp. 1355–1359, 2003.

Ramaswami, R.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, 2002).

Rosberg, Z.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance Analyses of Optical Burst-Switching Networks,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1187–1197.
[Crossref]

Santonja, V.

V. Santonja, “Dependability models of RAID using stochastic activity networks,” in Proceedings of Dependable Computing Conference, pp. 141–158, 1996.

Simeonidou, D.

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39(3) (2001) 128–135.
[Crossref]

Sivarajan, K. N.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, 2002).

Stol, N.

H. Øverby and N. Stol, “Effects of bursty traffic in service differentiated Optical Packet Switched networks,” Optics Express 12(3) (2004) 410–415.
[Crossref] [PubMed]

Stubkjaer, K. E.

S. L. Danielsen, C. Joergensen, B. Mikkelsen, and K. E. Stubkjaer, “Optical Packet Switched Network Layer Without Optical Buffers,” IEEE Photonics Technology Letters 10(6) (1998) 896–898.
[Crossref]

Tanenbaum, A. S.

A. S. Tanenbaum, Computer Networks (Prentice Hall, 1996).

Turner, J. S.

J. S. Turner, “Terabit burst switching,” Journal of High Speed Networks 8(1) (1999) 3–16.

Tzanakaki, A.

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39(3) (2001) 128–135.
[Crossref]

Vu, H. L.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance Analyses of Optical Burst-Switching Networks,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1187–1197.
[Crossref]

White, J.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance Analyses of Optical Burst-Switching Networks,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1187–1197.
[Crossref]

Wu, H.

Y. Chen, H. Wu, D. Xu, and C. Qiao, “Performance Analysis of Optical Burst Switched Node with Deflection Routing,” in Proceedings of International Conference on Communication, pp. 1355–1359, 2003.

Xu, D.

Y. Chen, H. Wu, D. Xu, and C. Qiao, “Performance Analysis of Optical Burst Switched Node with Deflection Routing,” in Proceedings of International Conference on Communication, pp. 1355–1359, 2003.

Yao, S.

S. Yao, B. Mukherjee, S. J. Ben Yoo, and S. Dixit, “A Unified Study of Contention-Resolution Schemes in Optical Packet-Switched Networks,” IEEE Journal of Lightwave Technology 21(3) (2003) 672–683.
[Crossref]

Yoo, M.

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP-Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18(10) (2000) 2062–2071.
[Crossref]

Yoo, S. J. Ben

S. Yao, B. Mukherjee, S. J. Ben Yoo, and S. Dixit, “A Unified Study of Contention-Resolution Schemes in Optical Packet-Switched Networks,” IEEE Journal of Lightwave Technology 21(3) (2003) 672–683.
[Crossref]

Zukerman, M.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance Analyses of Optical Burst-Switching Networks,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1187–1197.
[Crossref]

IEEE Communications Magazine (1)

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The Application of Optical Packet Switching in Future Communication Networks,” IEEE Communications Magazine 39(3) (2001) 128–135.
[Crossref]

IEEE Journal of Lightwave Technology (2)

D. K. Hunter, M. C. Chia, and I. Andonovic, “Buffering in Optical Packet Switches,” IEEE Journal of Lightwave Technology 16(12) (1998) 2081–2094.
[Crossref]

S. Yao, B. Mukherjee, S. J. Ben Yoo, and S. Dixit, “A Unified Study of Contention-Resolution Schemes in Optical Packet-Switched Networks,” IEEE Journal of Lightwave Technology 21(3) (2003) 672–683.
[Crossref]

IEEE Journal on Selected Areas in Communications (3)

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance Analyses of Optical Burst-Switching Networks,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1187–1197.
[Crossref]

L. Dittmann et al., “The European IST Project DAVID: A Viable Approach Toward Optical Packet Switching,” IEEE Journal on Selected Areas in Communications 21(7) (2003) 1026–1040.
[Crossref]

M. Yoo, C. Qiao, and S. Dixit, “QoS Performance of Optical Burst Switching in IP-Over-WDM Networks,” IEEE Journal on Selected Areas in Communications 18(10) (2000) 2062–2071.
[Crossref]

IEEE Photonics Technology Letters (1)

S. L. Danielsen, C. Joergensen, B. Mikkelsen, and K. E. Stubkjaer, “Optical Packet Switched Network Layer Without Optical Buffers,” IEEE Photonics Technology Letters 10(6) (1998) 896–898.
[Crossref]

Journal of High Speed Networks (1)

J. S. Turner, “Terabit burst switching,” Journal of High Speed Networks 8(1) (1999) 3–16.

Optics Express (1)

H. Øverby and N. Stol, “Effects of bursty traffic in service differentiated Optical Packet Switched networks,” Optics Express 12(3) (2004) 410–415.
[Crossref] [PubMed]

Other (10)

H. W. Braun, NLANR/Measurement and Network analysis http://www.caida.org/analysis/AIX/plen_hist/.

Y. Chen, H. Wu, D. Xu, and C. Qiao, “Performance Analysis of Optical Burst Switched Node with Deflection Routing,” in Proceedings of International Conference on Communication, pp. 1355–1359, 2003.

G. Birtwistle, DEMOS: Discrete Event Modelling on Simula (MacMillan, 1978).

E. V. Breusegem, J. Cheyns, B. Lannoo, A. Ackaert, M. Pickavet, and P. Demeester, “Implications of using offsets in all-optical packet switched networks,” in Proceedings of IEEE Optical Network Design and Modelling (Institute of Electrical and Electronic Engineers, 2003).

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, 2002).

A. S. Tanenbaum, Computer Networks (Prentice Hall, 1996).

V. Santonja, “Dependability models of RAID using stochastic activity networks,” in Proceedings of Dependable Computing Conference, pp. 141–158, 1996.

Fluid Studios, FSRaid documentation, http://www.fluidstudios.com/fsraid.html (accessed March 2004).

The Smart Par Primer, http://usenethelp.codeccorner.com/SPar_Primer.html (accessed March 2004).

Leonard Kleinrock, Queuing Systems Volume I: Theory (John Wiley & Sons, 1975).

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Figures (11)

Fig. 1.
Fig. 1.

Illustration of the NLPRS. One redundancy packet is created from a set of four data packets at ingress router j. All five packets are transmitted to egress router k. One data packet is dropped in the network due to contention. However, the lost data packet is reconstructed at egress router k using the successful data- and redundancy packet arrivals.

Fig. 2.
Fig. 2.

The considered optical packet switch and its corresponding electronic edge router.

Fig. 3.
Fig. 3.

The cumulative packet length distribution before the NLPRS has been applied (a). The cumulative end-to-end delay distribution before the NLPRS has been applied. ρD=0.2, G=7, AM=Poisson, PLD=Deterministic (b).

Fig. 4.
Fig. 4.

The DPLR as a function of the product r/m for various values of the parameter m. ρD=0.2, G=7, AM=Poisson, PLD=Deterministic, RPSM=BTB.

Fig. 5.
Fig. 5.

The DPLR as a function of the parameter ρD for various values of the parameter r. m=10, G=7, AM=Poisson, PLD=Deterministic, RPSM=BTB.

Fig. 6.
Fig. 6.

The DPLR as a function of the product r/m for various values of the parameter G. ρD=0.2, m=10, AM=Poisson, PLD=Deterministic, RPSM=BTB.

Fig. 7.
Fig. 7.

The DPLR as a function of the product r/m for various values of the parameter c. ρD=0.2, m=10, G=7, PLD=Deterministic, RPSM=BTB.

Fig. 8.
Fig. 8.

The DPLR as a function of the product r/m for various values of the parameter m. ρD=0.2, G=7, AM=Poisson, PLD=Empirically, RPSM=BTB.

Fig. 9.
Fig. 9.

The mean packet length as a function of the product r/m for various values of the parameter m. ρD=0.2, G=7, AM=Poisson, PLD=Empirically, RPSM=BTB (a). The cumulative packet length distribution when ρD=0.2, m=10, r=20, G=7, AM=Poisson, PLD=Empirically, RPSM=BTB (b).

Fig. 10.
Fig. 10.

The DPLR as a function of the product r/m for various redundancy packet scheduling mechanisms. ρD=0.2, m=5, G=7, AM=Poisson, PLD=Deterministic.

Fig. 11.
Fig. 11.

The cumulative end-to-end delay (in μs) when ρD=0.2, m=500, r=500, G=7, AM=Poisson, PLD=Empirically, RPSM=BTB (a). The cumulative end-to-end delay when ρD=0.4, m=10, r=10, G=7, AM=Poisson, PLD=Empirically, RPSM=BTB (b).

Tables (2)

Tables Icon

Table 1. The number of lost DPAR in a packet set as a function of the number of lost DPs and RPs.

Tables Icon

Table 2. The parameters used in the performance evaluation. Unless stated differently, the initial values of these parameters are used in the rest of the section.

Equations (20)

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

B i = E N i , A i = ( ( N i A i ) N i N i ! ) ( j = 0 N i ( N i A i ) j j ! )
A i = 1 μ N i ( π k e i e k λ k p = 1 C ( 1 I ( e p , e i , π k ) E N p , A p ) )
B ( π k ) = 1 e k ( 1 B i )
A i , r , m = 1 μ N i ( π k e i e k λ k r + m m p = 1 C ( 1 I ( e p , e i , π k ) E N p , A p , r , m ) )
= r + m m 1 μ N i ( π k e i e k λ k p = 1 C ( 1 I ( e p , e i , π k ) E N p , A p , r , m ) ) = r + m m A i
B i , r , m = E N i , A i , r , m = ( ( N i A i , r , m ) N i N i ! ) ( j = 0 N i ( N i A i , r , m ) j j ! )
B ( π k , r , m ) = 1 e k ( 1 B i , r , m )
Q s = m s B ( π k , r , m ) s ( 1 B ( π k , r , m ) ) m s
R s = r s B ( π k , r , m ) s ( 1 B ( π k , r , m ) ) r s
T m ( π k , r , m ) = i = 1 m j = Max [ r i + 1,0 ] r i Q i R j
= i = 1 m j = Max [ r i + 1,0 ] r i m i B ( π k , r , m ) i ( 1 B ( π k , r , m ) ) m i
· r j B ( π k , r , m ) j ( 1 B ( π k , r , m ) ) r j
= i = 1 m j = Max [ r i + 1,0 ] r i m i ( 1 e k ( 1 B i , r , m ) ) i ( e k ( 1 B i , r , m ) ) m i
· r j ( 1 e k ( 1 B i , r , m ) ) j ( e k ( 1 B i , r , m ) ) r j
T ( π k , r , m ) = 1 m T m ( π k , r , m )
RPLR NET = 1 λ TOT π k λ k B ( π k )
DPLR NET = 1 λ TOT π k λ k T ( π k , r , m )
ρ D = G γ · av · hop · L 3 GNC = γ · av · hop 3
a = ( 1 + ( c 2 1 ) c 2 + 1 ) 2 , θ 0 = 2 , θ 1 = 2 λ ( 1 a )
c 2 = ( σ m 1 ) 2 = m 2 m 1 2 1 = ε 1 = 2 a + k 2 a · k 2 ( a + k a · k ) 2 1

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