Abstract

We propose a technique for building optical packet networks that does not require any buffering, signaling or header processing. Contentions are solved by means of an optical device that allows the first packet to go through while blocking others. Blocked packets are redirected back to their source nodes, thus notifying the latter about the packet status. We describe the design principles of the corresponding all-optical networks, assess their performance and power consumption, and give examples of application in the context of access networks and data centers.

© 2013 Optical Society of America

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References

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  1. R. Tucker, “Scalability and energy consumption of optical and electronic packet switching,” J. Lightwave Technol., vol.  29, pp. 2410–2421, Aug. 2011.
    [CrossRef]
  2. K. Zhu and B. Mukherjee, “Traffic grooming in an optical WDM mesh network,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 122–133, Jan. 2002.
    [CrossRef]
  3. Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: a new area in optical networking research,” IEEE Network, vol.  18, no. 3, pp. 16–23, May-June 2004.
  4. M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The application of optical packet switching in future communication networks,” IEEE Commun. Mag., vol.  39, no. 3, pp. 128–135, Mar. 2001.
    [CrossRef]
  5. S. Di Lucente, N. Calabretta, J. A. C. Resing, and H. J. S. Dorren, “Scaling low-latency optical packet switches to a thousand ports,” in J. Opt. Commun. Netw., vol.  4, no. 9, pp. A17–A28, Sept. 2012.
    [CrossRef]
  6. B. Glance, “Protection-against-collision optical packet network,” J. Lightwave Technol., vol.  10, no. 9, pp. 1323–1328, Sept. 1992.
    [CrossRef]
  7. R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
    [CrossRef]
  8. T. Engset, “On the calculation of switches in an automatic telephone system,” in Tore Olaus Engset, A. Myskja and O. Espvik, Eds., 1998.
  9. C. Gallep and E. Conforti, “Reduction of semiconductor optical amplifier switching times by preimpulse step-injected current technique,” IEEE Photon. Technol. Lett., vol.  14, no. 7, pp. 902–904, July 2002.
    [CrossRef]
  10. T. Bonald, “Insensitive traffic models for communication networks,” Discrete Event Dyn. Syst., vol.  17, no. 3, pp. 405–421, Sept. 2007.
    [CrossRef]
  11. H. L. Vu, A. Zalesky, E. Wong, Z. Rosberg, S. Bilgrami, M. Zukerman, and R. Tucker, “Scalable performance evaluation of a hybrid optical switch,” J. Lightwave Technol., vol.  23, no. 10, pp. 2961–2973, Oct. 2005.
    [CrossRef]
  12. S. Ben Fredj, T. Bonald, A. Proutiere, G. Régnié, and J. W. Roberts, “Statistical bandwidth sharing: a study of congestion at flow level,” in Proc. ACM SIGCOMM, San Diego, CA, 2001.
  13. R. Serfozo, Introduction to Stochastic Networks. Springer, 1999.
  14. V. Eramo and M. Listanti, “Power consumption in bufferless optical packet switches in SOA technology,” J. Opt. Commun. Netw., vol.  1, no. 3, pp. B15–B29, Aug. 2009.
    [CrossRef]
  15. Xilinx 7 Series FPGA Power Benchmark Design Summary, Xilinx, San Jose, CA, July 2012, pp. 13–14.
  16. Cisco Systems, Cisco Nexus 5548 Data Sheets, San Jose, CA,2012.
  17. B. Ramamurthy, G. Rouskas, and K. Sivalingam, Next-Generation Internet: Architectures and Protocols. Cambridge Univ. Press, 2011.
  18. D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.
  19. T. Bonald, D. Cuda, R.-M. Indre, and L. Noirie, “Feasibility of optical switch-combiners,” LINCS, Tech. Rep., 2012 [Online]. Available: http://perso.telecom-paristech.fr/~bonald/Pub/doc.pdf .
  20. E. Sackinger, Broadband Circuits for Optical Fiber Communication. New York: Wiley, 2005, vol. 1.
  21. G. P. Agrawal, Fiber-Optic Communication Systems. New York: Wiley, 2002.
  22. Sun Telecom, SUN-GE8100 OLT Data Sheet, Johns Creek, GA, 2012.
  23. T. Benson, A. Akella, and D. Maltz, “Network traffic characteristics of data centers in the wild,” in Proc. IMC, Melbourne, Australia, 2010, pp. 267–280.

2012

S. Di Lucente, N. Calabretta, J. A. C. Resing, and H. J. S. Dorren, “Scaling low-latency optical packet switches to a thousand ports,” in J. Opt. Commun. Netw., vol.  4, no. 9, pp. A17–A28, Sept. 2012.
[CrossRef]

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

2011

2009

2007

T. Bonald, “Insensitive traffic models for communication networks,” Discrete Event Dyn. Syst., vol.  17, no. 3, pp. 405–421, Sept. 2007.
[CrossRef]

2005

2004

Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: a new area in optical networking research,” IEEE Network, vol.  18, no. 3, pp. 16–23, May-June 2004.

2002

K. Zhu and B. Mukherjee, “Traffic grooming in an optical WDM mesh network,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 122–133, Jan. 2002.
[CrossRef]

C. Gallep and E. Conforti, “Reduction of semiconductor optical amplifier switching times by preimpulse step-injected current technique,” IEEE Photon. Technol. Lett., vol.  14, no. 7, pp. 902–904, July 2002.
[CrossRef]

2001

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The application of optical packet switching in future communication networks,” IEEE Commun. Mag., vol.  39, no. 3, pp. 128–135, Mar. 2001.
[CrossRef]

1992

B. Glance, “Protection-against-collision optical packet network,” J. Lightwave Technol., vol.  10, no. 9, pp. 1323–1328, Sept. 1992.
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems. New York: Wiley, 2002.

Akella, A.

T. Benson, A. Akella, and D. Maltz, “Network traffic characteristics of data centers in the wild,” in Proc. IMC, Melbourne, Australia, 2010, pp. 267–280.

Akella, V.

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

Ben Fredj, S.

S. Ben Fredj, T. Bonald, A. Proutiere, G. Régnié, and J. W. Roberts, “Statistical bandwidth sharing: a study of congestion at flow level,” in Proc. ACM SIGCOMM, San Diego, CA, 2001.

Benson, T.

T. Benson, A. Akella, and D. Maltz, “Network traffic characteristics of data centers in the wild,” in Proc. IMC, Melbourne, Australia, 2010, pp. 267–280.

Bilgrami, S.

Bonald, T.

T. Bonald, “Insensitive traffic models for communication networks,” Discrete Event Dyn. Syst., vol.  17, no. 3, pp. 405–421, Sept. 2007.
[CrossRef]

S. Ben Fredj, T. Bonald, A. Proutiere, G. Régnié, and J. W. Roberts, “Statistical bandwidth sharing: a study of congestion at flow level,” in Proc. ACM SIGCOMM, San Diego, CA, 2001.

T. Bonald, D. Cuda, R.-M. Indre, and L. Noirie, “Feasibility of optical switch-combiners,” LINCS, Tech. Rep., 2012 [Online]. Available: http://perso.telecom-paristech.fr/~bonald/Pub/doc.pdf .

Calabretta, N.

Chen, Y.

Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: a new area in optical networking research,” IEEE Network, vol.  18, no. 3, pp. 16–23, May-June 2004.

Conforti, E.

C. Gallep and E. Conforti, “Reduction of semiconductor optical amplifier switching times by preimpulse step-injected current technique,” IEEE Photon. Technol. Lett., vol.  14, no. 7, pp. 902–904, July 2002.
[CrossRef]

Cuda, D.

T. Bonald, D. Cuda, R.-M. Indre, and L. Noirie, “Feasibility of optical switch-combiners,” LINCS, Tech. Rep., 2012 [Online]. Available: http://perso.telecom-paristech.fr/~bonald/Pub/doc.pdf .

D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.

Di Lucente, S.

Dorren, H. J. S.

Engset, T.

T. Engset, “On the calculation of switches in an automatic telephone system,” in Tore Olaus Engset, A. Myskja and O. Espvik, Eds., 1998.

Eramo, V.

Gallep, C.

C. Gallep and E. Conforti, “Reduction of semiconductor optical amplifier switching times by preimpulse step-injected current technique,” IEEE Photon. Technol. Lett., vol.  14, no. 7, pp. 902–904, July 2002.
[CrossRef]

Gaudino, R.

D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.

Gavilanes, G. A.

D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.

Glance, B.

B. Glance, “Protection-against-collision optical packet network,” J. Lightwave Technol., vol.  10, no. 9, pp. 1323–1328, Sept. 1992.
[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 Commun. Mag., vol.  39, no. 3, pp. 128–135, Mar. 2001.
[CrossRef]

Indre, R.-M.

T. Bonald, D. Cuda, R.-M. Indre, and L. Noirie, “Feasibility of optical switch-combiners,” LINCS, Tech. Rep., 2012 [Online]. Available: http://perso.telecom-paristech.fr/~bonald/Pub/doc.pdf .

Listanti, M.

Maier, G.

D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.

Maltz, D.

T. Benson, A. Akella, and D. Maltz, “Network traffic characteristics of data centers in the wild,” in Proc. IMC, Melbourne, Australia, 2010, pp. 267–280.

Mukherjee, B.

K. Zhu and B. Mukherjee, “Traffic grooming in an optical WDM mesh network,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 122–133, Jan. 2002.
[CrossRef]

Neri, F.

D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.

Nitta, C.

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

Noirie, L.

T. Bonald, D. Cuda, R.-M. Indre, and L. Noirie, “Feasibility of optical switch-combiners,” LINCS, Tech. Rep., 2012 [Online]. Available: http://perso.telecom-paristech.fr/~bonald/Pub/doc.pdf .

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 Commun. Mag., vol.  39, no. 3, pp. 128–135, Mar. 2001.
[CrossRef]

Proietti, R.

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

Proutiere, A.

S. Ben Fredj, T. Bonald, A. Proutiere, G. Régnié, and J. W. Roberts, “Statistical bandwidth sharing: a study of congestion at flow level,” in Proc. ACM SIGCOMM, San Diego, CA, 2001.

Qiao, C.

Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: a new area in optical networking research,” IEEE Network, vol.  18, no. 3, pp. 16–23, May-June 2004.

Raffaelli, C.

D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.

Ramamurthy, B.

B. Ramamurthy, G. Rouskas, and K. Sivalingam, Next-Generation Internet: Architectures and Protocols. Cambridge Univ. Press, 2011.

Régnié, G.

S. Ben Fredj, T. Bonald, A. Proutiere, G. Régnié, and J. W. Roberts, “Statistical bandwidth sharing: a study of congestion at flow level,” in Proc. ACM SIGCOMM, San Diego, CA, 2001.

Resing, J. A. C.

Roberts, J. W.

S. Ben Fredj, T. Bonald, A. Proutiere, G. Régnié, and J. W. Roberts, “Statistical bandwidth sharing: a study of congestion at flow level,” in Proc. ACM SIGCOMM, San Diego, CA, 2001.

Rosberg, Z.

Rouskas, G.

B. Ramamurthy, G. Rouskas, and K. Sivalingam, Next-Generation Internet: Architectures and Protocols. Cambridge Univ. Press, 2011.

Sackinger, E.

E. Sackinger, Broadband Circuits for Optical Fiber Communication. New York: Wiley, 2005, vol. 1.

Savi, M.

D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.

Serfozo, R.

R. Serfozo, Introduction to Stochastic Networks. Springer, 1999.

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 Commun. Mag., vol.  39, no. 3, pp. 128–135, Mar. 2001.
[CrossRef]

Sivalingam, K.

B. Ramamurthy, G. Rouskas, and K. Sivalingam, Next-Generation Internet: Architectures and Protocols. Cambridge Univ. Press, 2011.

Tucker, R.

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 Commun. Mag., vol.  39, no. 3, pp. 128–135, Mar. 2001.
[CrossRef]

Vu, H. L.

Wong, E.

Ye, X.

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

Yin, Y.

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

Yoo, S. J. B.

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

Yu, R.

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

Yu, X.

Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: a new area in optical networking research,” IEEE Network, vol.  18, no. 3, pp. 16–23, May-June 2004.

Zalesky, A.

Zhu, K.

K. Zhu and B. Mukherjee, “Traffic grooming in an optical WDM mesh network,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 122–133, Jan. 2002.
[CrossRef]

Zukerman, M.

Discrete Event Dyn. Syst.

T. Bonald, “Insensitive traffic models for communication networks,” Discrete Event Dyn. Syst., vol.  17, no. 3, pp. 405–421, Sept. 2007.
[CrossRef]

IEEE Commun. Mag.

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, “The application of optical packet switching in future communication networks,” IEEE Commun. Mag., vol.  39, no. 3, pp. 128–135, Mar. 2001.
[CrossRef]

IEEE J. Sel. Areas Commun.

K. Zhu and B. Mukherjee, “Traffic grooming in an optical WDM mesh network,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 122–133, Jan. 2002.
[CrossRef]

IEEE Network

Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: a new area in optical networking research,” IEEE Network, vol.  18, no. 3, pp. 16–23, May-June 2004.

IEEE Photon. Technol. Lett.

R. Proietti, Y. Yin, R. Yu, X. Ye, C. Nitta, V. Akella, and S. J. B. Yoo, “All-optical physical layer NACK in AWGR-based optical interconnects,” IEEE Photon. Technol. Lett., vol.  24, no. 5, pp. 410–412, Mar. 2012.
[CrossRef]

C. Gallep and E. Conforti, “Reduction of semiconductor optical amplifier switching times by preimpulse step-injected current technique,” IEEE Photon. Technol. Lett., vol.  14, no. 7, pp. 902–904, July 2002.
[CrossRef]

J. Lightwave Technol.

J. Opt. Commun. Netw.

Other

Xilinx 7 Series FPGA Power Benchmark Design Summary, Xilinx, San Jose, CA, July 2012, pp. 13–14.

Cisco Systems, Cisco Nexus 5548 Data Sheets, San Jose, CA,2012.

B. Ramamurthy, G. Rouskas, and K. Sivalingam, Next-Generation Internet: Architectures and Protocols. Cambridge Univ. Press, 2011.

D. Cuda, R. Gaudino, G. A. Gavilanes, F. Neri, G. Maier, C. Raffaelli, and M. Savi, “Capacity/cost tradeoffs in optical switching fabrics for terabit packet switches,” in Proc. ONDM, Braunschweig, Germany, Feb. 2009, pp. 1–6.

T. Bonald, D. Cuda, R.-M. Indre, and L. Noirie, “Feasibility of optical switch-combiners,” LINCS, Tech. Rep., 2012 [Online]. Available: http://perso.telecom-paristech.fr/~bonald/Pub/doc.pdf .

E. Sackinger, Broadband Circuits for Optical Fiber Communication. New York: Wiley, 2005, vol. 1.

G. P. Agrawal, Fiber-Optic Communication Systems. New York: Wiley, 2002.

Sun Telecom, SUN-GE8100 OLT Data Sheet, Johns Creek, GA, 2012.

T. Benson, A. Akella, and D. Maltz, “Network traffic characteristics of data centers in the wild,” in Proc. IMC, Melbourne, Australia, 2010, pp. 267–280.

T. Engset, “On the calculation of switches in an automatic telephone system,” in Tore Olaus Engset, A. Myskja and O. Espvik, Eds., 1998.

S. Ben Fredj, T. Bonald, A. Proutiere, G. Régnié, and J. W. Roberts, “Statistical bandwidth sharing: a study of congestion at flow level,” in Proc. ACM SIGCOMM, San Diego, CA, 2001.

R. Serfozo, Introduction to Stochastic Networks. Springer, 1999.

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Figures (13)

Fig. 1.
Fig. 1.

N×1 DOC. The continuous lines represent optical signals while the dashed-dotted lines correspond to electrical signals.

Fig. 2.
Fig. 2.

Source node able to detect the feedback received from a DOC. Continuous and dotted lines represent optical and electrical signals, respectively.

Fig. 3.
Fig. 3.

2×2 optical switch-combiner with W=4 wavelengths per fiber.

Fig. 4.
Fig. 4.

Utilization of the channel by homogeneous sources for mean normalized backoff times b=1 (top curves) and b=10 (bottom curves).

Fig. 5.
Fig. 5.

Utilization of the channel by two types of sources with mean normalized backoff times b1=1 (top curves) and b2=10 (bottom curves).

Fig. 6.
Fig. 6.

Flow throughput at the output of a switch-combiner with N=100 input ports (b=1, 5, 10 from top to bottom).

Fig. 7.
Fig. 7.

Maximum utilization with respect to the number of ports.

Fig. 8.
Fig. 8.

Flow throughput at the output of a switch-combiner with N=8 input ports (b=1, 5, 10 from top to bottom), each input port connecting a switch-combiner that aggregates the traffic of 100 edge nodes.

Fig. 9.
Fig. 9.

Flow throughput due to the transmitter constraint (b=1, 5, 10 from top to bottom).

Fig. 10.
Fig. 10.

Probability that packet delay exceeds 1 ms for a single switch-combiner with N=100 input ports (b=10, 1 from top to bottom).

Fig. 11.
Fig. 11.

Probability that packet delay exceeds 1 ms for a switch-combiner with N=8 input ports (b=10, 1 from top to bottom), each input port connecting a switch-combiner that aggregates the traffic of 100 edge nodes.

Fig. 12.
Fig. 12.

Average power consumption of a DOC due to the SOAs and RSOAs.

Fig. 13.
Fig. 13.

An all-optical data center.

Tables (1)

Tables Icon

TABLE I Maximum Number of Ports of the DOC

Equations (23)

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

U(n)=nνnν+μ=nn+b,
B=n1n1+b.
U(n)=i=1nvii=1nvi+μ=nn+m,
m=(1ni=1n1bi)1,
Bi=jiνjjiνj+μ=ji1bjji1bj+1.
b˜=b+B˜,
B˜=n1n1+b˜.
b˜=12((n+b+1)24(b+1)n+b+1).
U(n)=nn+b˜.
b˜i=bi+B˜i,
B˜i=ji1b˜jji1b˜j+1.
U(n)=nn+m˜,
m˜=(1ni=1n1b˜i)1.
ρ=λσR.
γ=ρRE(n).
γ=R(1ρ).
γRb+1(1ρ).
π(n)=π(0)ρnU(1)U(n).
γRb+1.
π(n)=π(0)(n+bn)ρn.
γ=Rb+1(1ρ).
U(N)=NN+12((N+1)24N+1).
P=P(R)SOANρN(1ρ)+ρ.