Abstract

This paper proposes three-domain (3-D) burst scheduling in optically burst-switched WDM networks. This scheme completely eliminates contention at intermediate core nodes by combining contention avoidance schemes in the space, wavelength, and time domains. In the space and wavelength domains, the proposed scheme constructs multiple fixed, tree-shaped routes for burst transmission called λ-trees, each of which is assigned a wavelength. Note that λ-trees with the same wavelength do not share any links, and therefore the proposed scheme provides contention-free transmission among bursts transmitted on different λ-trees. Furthermore, in the time domain, the proposed scheme introduces an ingenious offset time assignment scheme named burst grooming, which eliminates contention on each λ-tree. As a result, contention at intermediate core nodes is completely eliminated because each ingress node schedules incoming bursts onto λ-trees with burst grooming. We provide λ-tree construction and burst scheduling algorithms, which aim to minimize the overall burst loss probability. Extensive simulation experiments show that the proposed scheme improves the overall burst loss performance dramatically without wavelength conversion.

© 2010 Optical Society of America

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  1. C. Qiao, M. Yoo, “Optical burst switching (OBS): a new paradigm for an optical internet,” J. High Speed Netw., vol. 8, no. 1, pp. 69–84, 1999.
  2. S. Yao, B. Mukherjee, S. Dixit, “A unified study of contention-resolution schemes in optical packet-switched networks,” J. Lightwave Technol., vol. 21, no. 3, pp. 672–83, 2003.
    [CrossRef]
  3. J. Li, C. Qiao, “Schedule burst proactively for optical burst switched networks,” Comput. Netw., vol. 44, no. 5, pp. 617–629, 2004.
    [CrossRef]
  4. J. Pedro, P. Monteiro, J. Pires, “Wavelength contention minimization strategies for optical burst-switched networks,” in IEEE Global Telecommunications Conf. (GLOBECOM 2006), San Francisco, CA, 2006.
    [CrossRef]
  5. J. Teng, G. Rouskas, “Wavelength selection in OBS networks using traffic engineering and priority-based concept,” IEEE J. Sel. Areas Commun., vol. 23, no. 8, pp. 1658–1669, 2005.
    [CrossRef]
  6. J. Turner, “Terabit burst switching,” J. High Speed Netw., vol. 8, no. 1, pp. 3–16, 1999.
  7. X. Wang, H. Morikawa, T. Aoyama, “Priority-based wavelength assignment algorithm for burst switched photonic networks,” in Optical Fiber Communication Conf., Anaheim, CA, 2002, pp. 765–767.
  8. Y. Xiong, M. Vandenhoute, H. C. Cankaya, “Control architecture in optical burst-switched WDM networks,” IEEE J. Sel. Areas Commun., vol. 8, no. 10, pp. 1838–1851, 2000.
    [CrossRef]
  9. J. Xu, C. Qiao, J. Li, G. Xu, “Efficient channel scheduling algorithms in optical-burst-switched networks,” in IEEE Conf. on Computer Communications (INFOCOM), Orlando, FL, 2003, pp. 2268–2278.
  10. K. Hirata, T. Matsuda, H. Nagamochi, T. Takine, “Contention-free λ-planes in optically burst switched WDM networks,” IEICE Trans. Commun., vol. E90-E, no. 9, pp. 2524–2531, 2007.
    [CrossRef]
  11. Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
    [CrossRef]
  12. H. Zung, J. P. Jue, B. Mukherjee, “A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks,” Opt. Networks Mag., vol. 1, no. 1, pp. 47–60, 2000.
  13. C. H. Papadimitriou, K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Mineola, NY: Dover Publications, 1998.
  14. A. Nucci, A. Sridharan, N. Taft, “The problem of synthetically generating IP traffic matrices: initial recommendations,” Comput. Commun. Rev., vol. 35, no. 3, pp. 19–32, 2005.
    [CrossRef]
  15. Y. Du, T. Pu, H. Zhang, Y. Quo, “Adaptive load balancing routing algorithm for optical burst-switching networks,” in Optical Fiber Communication Conf., Anaheim, CA, Mar. 2006, paper OThF7.
  16. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms. Cambridge, MA: MIT Press, 2001.

2007 (1)

K. Hirata, T. Matsuda, H. Nagamochi, T. Takine, “Contention-free λ-planes in optically burst switched WDM networks,” IEICE Trans. Commun., vol. E90-E, no. 9, pp. 2524–2531, 2007.
[CrossRef]

2006 (1)

Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
[CrossRef]

2005 (2)

A. Nucci, A. Sridharan, N. Taft, “The problem of synthetically generating IP traffic matrices: initial recommendations,” Comput. Commun. Rev., vol. 35, no. 3, pp. 19–32, 2005.
[CrossRef]

J. Teng, G. Rouskas, “Wavelength selection in OBS networks using traffic engineering and priority-based concept,” IEEE J. Sel. Areas Commun., vol. 23, no. 8, pp. 1658–1669, 2005.
[CrossRef]

2004 (1)

J. Li, C. Qiao, “Schedule burst proactively for optical burst switched networks,” Comput. Netw., vol. 44, no. 5, pp. 617–629, 2004.
[CrossRef]

2003 (1)

2000 (2)

Y. Xiong, M. Vandenhoute, H. C. Cankaya, “Control architecture in optical burst-switched WDM networks,” IEEE J. Sel. Areas Commun., vol. 8, no. 10, pp. 1838–1851, 2000.
[CrossRef]

H. Zung, J. P. Jue, B. Mukherjee, “A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks,” Opt. Networks Mag., vol. 1, no. 1, pp. 47–60, 2000.

1999 (2)

C. Qiao, M. Yoo, “Optical burst switching (OBS): a new paradigm for an optical internet,” J. High Speed Netw., vol. 8, no. 1, pp. 69–84, 1999.

J. Turner, “Terabit burst switching,” J. High Speed Netw., vol. 8, no. 1, pp. 3–16, 1999.

Abe, M.

Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
[CrossRef]

Aoyama, T.

X. Wang, H. Morikawa, T. Aoyama, “Priority-based wavelength assignment algorithm for burst switched photonic networks,” in Optical Fiber Communication Conf., Anaheim, CA, 2002, pp. 765–767.

Cankaya, H. C.

Y. Xiong, M. Vandenhoute, H. C. Cankaya, “Control architecture in optical burst-switched WDM networks,” IEEE J. Sel. Areas Commun., vol. 8, no. 10, pp. 1838–1851, 2000.
[CrossRef]

Cormen, T. H.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms. Cambridge, MA: MIT Press, 2001.

Dixit, S.

Du, Y.

Y. Du, T. Pu, H. Zhang, Y. Quo, “Adaptive load balancing routing algorithm for optical burst-switching networks,” in Optical Fiber Communication Conf., Anaheim, CA, Mar. 2006, paper OThF7.

Hirata, K.

K. Hirata, T. Matsuda, H. Nagamochi, T. Takine, “Contention-free λ-planes in optically burst switched WDM networks,” IEICE Trans. Commun., vol. E90-E, no. 9, pp. 2524–2531, 2007.
[CrossRef]

Jue, J. P.

H. Zung, J. P. Jue, B. Mukherjee, “A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks,” Opt. Networks Mag., vol. 1, no. 1, pp. 47–60, 2000.

Leiserson, C. E.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms. Cambridge, MA: MIT Press, 2001.

Li, J.

J. Li, C. Qiao, “Schedule burst proactively for optical burst switched networks,” Comput. Netw., vol. 44, no. 5, pp. 617–629, 2004.
[CrossRef]

J. Xu, C. Qiao, J. Li, G. Xu, “Efficient channel scheduling algorithms in optical-burst-switched networks,” in IEEE Conf. on Computer Communications (INFOCOM), Orlando, FL, 2003, pp. 2268–2278.

Masuda, H.

Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
[CrossRef]

Matsuda, T.

K. Hirata, T. Matsuda, H. Nagamochi, T. Takine, “Contention-free λ-planes in optically burst switched WDM networks,” IEICE Trans. Commun., vol. E90-E, no. 9, pp. 2524–2531, 2007.
[CrossRef]

Miyagawa, Y.

Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
[CrossRef]

Monteiro, P.

J. Pedro, P. Monteiro, J. Pires, “Wavelength contention minimization strategies for optical burst-switched networks,” in IEEE Global Telecommunications Conf. (GLOBECOM 2006), San Francisco, CA, 2006.
[CrossRef]

Morikawa, H.

X. Wang, H. Morikawa, T. Aoyama, “Priority-based wavelength assignment algorithm for burst switched photonic networks,” in Optical Fiber Communication Conf., Anaheim, CA, 2002, pp. 765–767.

Mukherjee, B.

S. Yao, B. Mukherjee, S. Dixit, “A unified study of contention-resolution schemes in optical packet-switched networks,” J. Lightwave Technol., vol. 21, no. 3, pp. 672–83, 2003.
[CrossRef]

H. Zung, J. P. Jue, B. Mukherjee, “A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks,” Opt. Networks Mag., vol. 1, no. 1, pp. 47–60, 2000.

Nagamochi, H.

K. Hirata, T. Matsuda, H. Nagamochi, T. Takine, “Contention-free λ-planes in optically burst switched WDM networks,” IEICE Trans. Commun., vol. E90-E, no. 9, pp. 2524–2531, 2007.
[CrossRef]

Nucci, A.

A. Nucci, A. Sridharan, N. Taft, “The problem of synthetically generating IP traffic matrices: initial recommendations,” Comput. Commun. Rev., vol. 35, no. 3, pp. 19–32, 2005.
[CrossRef]

Papadimitriou, C. H.

C. H. Papadimitriou, K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Mineola, NY: Dover Publications, 1998.

Pedro, J.

J. Pedro, P. Monteiro, J. Pires, “Wavelength contention minimization strategies for optical burst-switched networks,” in IEEE Global Telecommunications Conf. (GLOBECOM 2006), San Francisco, CA, 2006.
[CrossRef]

Pires, J.

J. Pedro, P. Monteiro, J. Pires, “Wavelength contention minimization strategies for optical burst-switched networks,” in IEEE Global Telecommunications Conf. (GLOBECOM 2006), San Francisco, CA, 2006.
[CrossRef]

Pu, T.

Y. Du, T. Pu, H. Zhang, Y. Quo, “Adaptive load balancing routing algorithm for optical burst-switching networks,” in Optical Fiber Communication Conf., Anaheim, CA, Mar. 2006, paper OThF7.

Qiao, C.

J. Li, C. Qiao, “Schedule burst proactively for optical burst switched networks,” Comput. Netw., vol. 44, no. 5, pp. 617–629, 2004.
[CrossRef]

C. Qiao, M. Yoo, “Optical burst switching (OBS): a new paradigm for an optical internet,” J. High Speed Netw., vol. 8, no. 1, pp. 69–84, 1999.

J. Xu, C. Qiao, J. Li, G. Xu, “Efficient channel scheduling algorithms in optical-burst-switched networks,” in IEEE Conf. on Computer Communications (INFOCOM), Orlando, FL, 2003, pp. 2268–2278.

Quo, Y.

Y. Du, T. Pu, H. Zhang, Y. Quo, “Adaptive load balancing routing algorithm for optical burst-switching networks,” in Optical Fiber Communication Conf., Anaheim, CA, Mar. 2006, paper OThF7.

Rivest, R. L.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms. Cambridge, MA: MIT Press, 2001.

Rouskas, G.

J. Teng, G. Rouskas, “Wavelength selection in OBS networks using traffic engineering and priority-based concept,” IEEE J. Sel. Areas Commun., vol. 23, no. 8, pp. 1658–1669, 2005.
[CrossRef]

Sridharan, A.

A. Nucci, A. Sridharan, N. Taft, “The problem of synthetically generating IP traffic matrices: initial recommendations,” Comput. Commun. Rev., vol. 35, no. 3, pp. 19–32, 2005.
[CrossRef]

Steiglitz, K.

C. H. Papadimitriou, K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Mineola, NY: Dover Publications, 1998.

Stein, C.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms. Cambridge, MA: MIT Press, 2001.

Taft, N.

A. Nucci, A. Sridharan, N. Taft, “The problem of synthetically generating IP traffic matrices: initial recommendations,” Comput. Commun. Rev., vol. 35, no. 3, pp. 19–32, 2005.
[CrossRef]

Takahashi, H.

Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
[CrossRef]

Takara, H.

Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
[CrossRef]

Takine, T.

K. Hirata, T. Matsuda, H. Nagamochi, T. Takine, “Contention-free λ-planes in optically burst switched WDM networks,” IEICE Trans. Commun., vol. E90-E, no. 9, pp. 2524–2531, 2007.
[CrossRef]

Teng, J.

J. Teng, G. Rouskas, “Wavelength selection in OBS networks using traffic engineering and priority-based concept,” IEEE J. Sel. Areas Commun., vol. 23, no. 8, pp. 1658–1669, 2005.
[CrossRef]

Turner, J.

J. Turner, “Terabit burst switching,” J. High Speed Netw., vol. 8, no. 1, pp. 3–16, 1999.

Vandenhoute, M.

Y. Xiong, M. Vandenhoute, H. C. Cankaya, “Control architecture in optical burst-switched WDM networks,” IEEE J. Sel. Areas Commun., vol. 8, no. 10, pp. 1838–1851, 2000.
[CrossRef]

Wang, X.

X. Wang, H. Morikawa, T. Aoyama, “Priority-based wavelength assignment algorithm for burst switched photonic networks,” in Optical Fiber Communication Conf., Anaheim, CA, 2002, pp. 765–767.

Xiong, Y.

Y. Xiong, M. Vandenhoute, H. C. Cankaya, “Control architecture in optical burst-switched WDM networks,” IEEE J. Sel. Areas Commun., vol. 8, no. 10, pp. 1838–1851, 2000.
[CrossRef]

Xu, G.

J. Xu, C. Qiao, J. Li, G. Xu, “Efficient channel scheduling algorithms in optical-burst-switched networks,” in IEEE Conf. on Computer Communications (INFOCOM), Orlando, FL, 2003, pp. 2268–2278.

Xu, J.

J. Xu, C. Qiao, J. Li, G. Xu, “Efficient channel scheduling algorithms in optical-burst-switched networks,” in IEEE Conf. on Computer Communications (INFOCOM), Orlando, FL, 2003, pp. 2268–2278.

Yamamoto, T.

Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
[CrossRef]

Yao, S.

Yoo, M.

C. Qiao, M. Yoo, “Optical burst switching (OBS): a new paradigm for an optical internet,” J. High Speed Netw., vol. 8, no. 1, pp. 69–84, 1999.

Zhang, H.

Y. Du, T. Pu, H. Zhang, Y. Quo, “Adaptive load balancing routing algorithm for optical burst-switching networks,” in Optical Fiber Communication Conf., Anaheim, CA, Mar. 2006, paper OThF7.

Zung, H.

H. Zung, J. P. Jue, B. Mukherjee, “A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks,” Opt. Networks Mag., vol. 1, no. 1, pp. 47–60, 2000.

Comput. Commun. Rev. (1)

A. Nucci, A. Sridharan, N. Taft, “The problem of synthetically generating IP traffic matrices: initial recommendations,” Comput. Commun. Rev., vol. 35, no. 3, pp. 19–32, 2005.
[CrossRef]

Comput. Netw. (1)

J. Li, C. Qiao, “Schedule burst proactively for optical burst switched networks,” Comput. Netw., vol. 44, no. 5, pp. 617–629, 2004.
[CrossRef]

IEEE J. Sel. Areas Commun. (2)

J. Teng, G. Rouskas, “Wavelength selection in OBS networks using traffic engineering and priority-based concept,” IEEE J. Sel. Areas Commun., vol. 23, no. 8, pp. 1658–1669, 2005.
[CrossRef]

Y. Xiong, M. Vandenhoute, H. C. Cankaya, “Control architecture in optical burst-switched WDM networks,” IEEE J. Sel. Areas Commun., vol. 8, no. 10, pp. 1838–1851, 2000.
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, H. Takara, “Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source,” IEEE Photon. Technol. Lett., vol. 42, no. 11, pp. 655–657, 2006.
[CrossRef]

IEICE Trans. Commun. (1)

K. Hirata, T. Matsuda, H. Nagamochi, T. Takine, “Contention-free λ-planes in optically burst switched WDM networks,” IEICE Trans. Commun., vol. E90-E, no. 9, pp. 2524–2531, 2007.
[CrossRef]

J. High Speed Netw. (2)

C. Qiao, M. Yoo, “Optical burst switching (OBS): a new paradigm for an optical internet,” J. High Speed Netw., vol. 8, no. 1, pp. 69–84, 1999.

J. Turner, “Terabit burst switching,” J. High Speed Netw., vol. 8, no. 1, pp. 3–16, 1999.

J. Lightwave Technol. (1)

Opt. Networks Mag. (1)

H. Zung, J. P. Jue, B. Mukherjee, “A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks,” Opt. Networks Mag., vol. 1, no. 1, pp. 47–60, 2000.

Other (6)

C. H. Papadimitriou, K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Mineola, NY: Dover Publications, 1998.

J. Xu, C. Qiao, J. Li, G. Xu, “Efficient channel scheduling algorithms in optical-burst-switched networks,” in IEEE Conf. on Computer Communications (INFOCOM), Orlando, FL, 2003, pp. 2268–2278.

X. Wang, H. Morikawa, T. Aoyama, “Priority-based wavelength assignment algorithm for burst switched photonic networks,” in Optical Fiber Communication Conf., Anaheim, CA, 2002, pp. 765–767.

J. Pedro, P. Monteiro, J. Pires, “Wavelength contention minimization strategies for optical burst-switched networks,” in IEEE Global Telecommunications Conf. (GLOBECOM 2006), San Francisco, CA, 2006.
[CrossRef]

Y. Du, T. Pu, H. Zhang, Y. Quo, “Adaptive load balancing routing algorithm for optical burst-switching networks,” in Optical Fiber Communication Conf., Anaheim, CA, Mar. 2006, paper OThF7.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms. Cambridge, MA: MIT Press, 2001.

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Figures (16)

Fig. 1
Fig. 1

Primary and secondary λ-trees.

Fig. 2
Fig. 2

Burst grooming on a λ-tree.

Fig. 3
Fig. 3

Effect of offset times.

Fig. 4
Fig. 4

Burst grooming (R and L denote the root and the leaf, respectively).

Fig. 5
Fig. 5

Network model.

Fig. 6
Fig. 6

Burst loss probability as a function of the offered load ( K = 10 , W = 64 ).

Fig. 7
Fig. 7

Comparison of primary trees in terms of burst loss probability ( K = 10 , W = 64 ).

Fig. 8
Fig. 8

Burst loss probability as a function of the offered load ( K = 10 , W = 16 ).

Fig. 9
Fig. 9

Burst loss probability as a function of the offered load ( K = 10 , W = 32 ).

Fig. 10
Fig. 10

Burst loss probability as a function of the offered load ( K = 10 , W = 64 ).

Fig. 11
Fig. 11

Burst loss probability as a function of the number of wavelengths ( K = 10 , ρ = 0.55 ).

Fig. 12
Fig. 12

Average end-to-end delay as a function of the offered load ( K = 10 , W = 64 ).

Fig. 13
Fig. 13

Burst loss probability as a function of K ( ρ = 0.55 , W = 64 ).

Fig. 14
Fig. 14

Average end-to-end delay as a function of ( ρ = 0.05 , W = 64 ).

Fig. 15
Fig. 15

Candidate partial trees.

Fig. 16
Fig. 16

An example of D k ( T i , e , w ) (R denotes the root ingress node).

Tables (2)

Tables Icon

Table 1 Traffic Matrix

Tables Icon

Table 2 List of Symbols Used in Our Algorithms

Equations (9)

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

offset time H × Δ ,
offset time = L max × ( H max 1 ) + H max × Δ ,
ρ = i I j J λ i , j L h i , j total # of wavelengths × total # of links ,
Λ i n , j = P i n , j λ i n , j .
R i , e , w = j J i , e , w Λ i , j .
Λ i * , j P i * , j Λ i * , j .
Λ i , j P i , j Λ i , j .
Λ i l , i k , p k + = m D i k , p k ( T i 0 , e , w ) Λ i l , m , 0 l k n ,
P i k , j = l = 0 k 1 Λ i l , i k , p k + E [ L i l ] ( 1 P i l , j ) + [ 1 l = 0 k 1 Λ i l , i k , p k + E [ L i l ] ( 1 P i l , j ) ] [ 1 L i k * ( l = 0 k 1 Λ i l , i k , p k + ( 1 P i l , j ) ) ] + [ 1 l = 0 k 1 Λ i l , i k , p k + E [ L i l ] ( 1 P i l , j ) ] L i k * [ l = 0 k 1 Λ i l , i k , p k + ( 1 P i l , j ) ] Λ i k , i k , p k + E [ L i k ] 1 + Λ i k , i k , p k + E [ L i k ] = 1 [ 1 l = 0 k 1 Λ i l , i k , p k + E [ L i l ] ( 1 P i l , j ) ] L i k * [ l = 0 k 1 Λ i l , i k , p k + ( 1 P i l , j ) ] 1 + Λ i k , i k , p k + E [ L i k ] ,