Abstract

For decades, optical networks have provided larger bandwidths than could be utilized, but with the increasing growth of the global Internet traffic demand, new optical transmission technologies are required to provide a much higher data rate per channel and to enable more flexibility in the allocation of traffic flow. Currently, researchers are investigating innovative transceiver architectures capable of dynamically adapting the modulation format to the transmission link properties. These transceivers are referred to as elastic and enable flexible allocation of optical bandwidth resources. To exploit their capabilities, the conventional fixed spectrum grid has to evolve to provide a more scalable and flexible system that can provide the spectral resources requireded to serve the client demand. The benefits of elastic transceivers with distance-adaptive data rates have been evaluated in optical core networks, but their application to metro ring networks has still not been addressed. This paper proposes methods based on integer linear programs and heuristic approaches to solve the routing, modulation level, and spectrum assignment problem in optical rings with elastic transceivers and rate-adaptive modulation formats. Moreover, we discuss how to analytically compute feasible solutions that provide useful upper bounds. Results show a significant reduction in terms of transceiver utilization and spectrum occupation.

© 2013 Optical Society of America

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2013

C. Rottondi, M. Tornatore, A. Pattavina, and G. Gavioli, “Traffic grooming and spectrum assignment for coherent transceivers in metro-flexible networks,” IEEE Photon. Technol. Lett., vol.  25, pp. 183–186, Jan. 2013.
[CrossRef]

2012

G. Shen and M. Zukerman, “Spectrum-efficient and agile CO-OFDM optical transport networks: architecture, design, and operation,” IEEE Commun. Mag., vol.  50, no. 5, pp. 82–89, May 2012.
[CrossRef]

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–204, July 2012.
[CrossRef]

F. Cugini, G. Meloni, F. Paolucci, N. Sambo, M. Secondini, L. Gerardi, L. Potì, and P. Castoldi, “Demonstration of flexible optical network based on path computation element,” J. Lightwave Technol., vol.  30, no. 5, pp. 727–733, Mar. 2012.
[CrossRef]

2011

2010

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

2009

W. Wei, J. Hu, D. Qian, P. Ji, T. Wang, X. Liu, and C. Qiao, “PONIARD: a programmable optical networking infrastructure for advanced research and development of future Internet,” J. Lightwave Technol., vol.  27, no. 3, pp. 233–242, 2009.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, Nov. 2009.
[CrossRef]

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

2002

R. Dutta and G. Rouskas, “On optimal traffic grooming in WDM rings,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 110–121, Jan. 2002.
[CrossRef]

2000

O. Gerstel, R. Ramaswami, and G. Sasaki, “Cost-effective traffic grooming in WDM rings,” IEEE/ACM Trans. Netw., vol.  8, no. 5, pp. 618–630, Oct. 2000.
[CrossRef]

Alfiad, M. S.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Amaya, N.

N. Amaya, I. Muhammad, G. S. Zervas, R. Nejabati, D. Simeonidou, Y. Zhou, and A. Lord, “Experimental demonstration of a gridless multi-granular optical network supporting flexible spectrum switching,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Antona, J.-C.

O. Rival, A. Morea, and J.-C. Antona, “Optical network planning with rate-tunable NRZ transponders,” in Proc. ECOC, Vienna, Austria, Sept. 2009.

Basch, B.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Behel, J.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Bosco, G.

Cao, X.

Y. Wang, X. Cao, and Q. Hu, “Routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in Proc. IEEE ICC, Kyoto, Japan, June 2011.

Y. Wang, X. Cao, and Y. Pan, “A study of the routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in Proc. IEEE INFOCOM, Shangai, China, 2011.

Carena, A.

Castoldi, P.

Chbat, M.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Chen, D.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Christodoulopoulos, K.

Cugini, F.

Curri, V.

de Man, E.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

de Waardt, H.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Dutta, R.

R. Dutta and G. Rouskas, “On optimal traffic grooming in WDM rings,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 110–121, Jan. 2002.
[CrossRef]

Finkenzeller, M.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Forghieri, F.

Freiberger, M.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Gavioli, G.

C. Rottondi, M. Tornatore, A. Pattavina, and G. Gavioli, “Traffic grooming and spectrum assignment for coherent transceivers in metro-flexible networks,” IEEE Photon. Technol. Lett., vol.  25, pp. 183–186, Jan. 2013.
[CrossRef]

C. Rottondi, M. Tornatore, F. Puleio, S. Raavi, A. Pattavina, and G. Gavioli, “On the benefits of elastic transponders in optical metro networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2012, paper OW3A.6.

Gerardi, L.

Gerstel, O.

O. Gerstel, R. Ramaswami, and G. Sasaki, “Cost-effective traffic grooming in WDM rings,” IEEE/ACM Trans. Netw., vol.  8, no. 5, pp. 618–630, Oct. 2000.
[CrossRef]

Guo, W.

W. Zheng, Y. Jin, W. Sun, W. Guo, and W. Hu, “On the spectrum-efficiency of bandwidth-variable optical OFDM transport networks,” in Optical Fiber Communication Conf., San Diego, CA, Mar. 2010, paper OWR5.

Hirano, A.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

Hu, J.

Hu, Q.

Y. Wang, X. Cao, and Q. Hu, “Routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in Proc. IEEE ICC, Kyoto, Japan, June 2011.

Hu, W.

W. Zheng, Y. Jin, W. Sun, W. Guo, and W. Hu, “On the spectrum-efficiency of bandwidth-variable optical OFDM transport networks,” in Optical Fiber Communication Conf., San Diego, CA, Mar. 2010, paper OWR5.

Hua, N.

Y. Zhang, X. Zheng, Q. Li, N. Hua, Y. Li, and H. Zhang, “Traffic grooming in spectrum-elastic optical path networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI1.

Ji, P.

Ji, P. N.

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–204, July 2012.
[CrossRef]

Jin, Y.

W. Zheng, Y. Jin, W. Sun, W. Guo, and W. Hu, “On the spectrum-efficiency of bandwidth-variable optical OFDM transport networks,” in Optical Fiber Communication Conf., San Diego, CA, Mar. 2010, paper OWR5.

Jinno, M.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, Nov. 2009.
[CrossRef]

Jue, J. P.

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–204, July 2012.
[CrossRef]

Kozicki, B.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, Nov. 2009.
[CrossRef]

Kuschnerov, M.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Lankl, B.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Lee, W.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Li, Q.

Y. Zhang, X. Zheng, Q. Li, N. Hua, Y. Li, and H. Zhang, “Traffic grooming in spectrum-elastic optical path networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI1.

Li, Y.

Y. Zhang, X. Zheng, Q. Li, N. Hua, Y. Li, and H. Zhang, “Traffic grooming in spectrum-elastic optical path networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI1.

Liu, X.

Lord, A.

N. Amaya, I. Muhammad, G. S. Zervas, R. Nejabati, D. Simeonidou, Y. Zhou, and A. Lord, “Experimental demonstration of a gridless multi-granular optical network supporting flexible spectrum switching,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Matsuoka, S.

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, Nov. 2009.
[CrossRef]

Meloni, G.

More, A.

O. Rival and A. More, “Cost-efficiency of mixed 10-40-100  Gb/s networks and elastic optical networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI4.

Morea, A.

O. Rival, A. Morea, and J.-C. Antona, “Optical network planning with rate-tunable NRZ transponders,” in Proc. ECOC, Vienna, Austria, Sept. 2009.

Muhammad, I.

N. Amaya, I. Muhammad, G. S. Zervas, R. Nejabati, D. Simeonidou, Y. Zhou, and A. Lord, “Experimental demonstration of a gridless multi-granular optical network supporting flexible spectrum switching,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Mukherjee, B.

B. Mukherjee, Optical WDM Networks.Springer, 2006.

R. Wang and B. Mukherjee, “Spectrum management in heterogeneous bandwidth networks,” in IEEE GLOBECOM, Anaheim, CA, Dec. 2012.

Napoli, A.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Nejabati, R.

N. Amaya, I. Muhammad, G. S. Zervas, R. Nejabati, D. Simeonidou, Y. Zhou, and A. Lord, “Experimental demonstration of a gridless multi-granular optical network supporting flexible spectrum switching,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Pan, Y.

Y. Wang, X. Cao, and Y. Pan, “A study of the routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in Proc. IEEE INFOCOM, Shangai, China, 2011.

Paolucci, F.

Patel, A. N.

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–204, July 2012.
[CrossRef]

Pattavina, A.

C. Rottondi, M. Tornatore, A. Pattavina, and G. Gavioli, “Traffic grooming and spectrum assignment for coherent transceivers in metro-flexible networks,” IEEE Photon. Technol. Lett., vol.  25, pp. 183–186, Jan. 2013.
[CrossRef]

C. Rottondi, M. Tornatore, F. Puleio, S. Raavi, A. Pattavina, and G. Gavioli, “On the benefits of elastic transponders in optical metro networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2012, paper OW3A.6.

Peterson, D.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Poggiolini, P.

Pollock, M.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Potì, L.

Puleio, F.

C. Rottondi, M. Tornatore, F. Puleio, S. Raavi, A. Pattavina, and G. Gavioli, “On the benefits of elastic transponders in optical metro networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2012, paper OW3A.6.

Qian, D.

Qiao, C.

Raavi, S.

C. Rottondi, M. Tornatore, F. Puleio, S. Raavi, A. Pattavina, and G. Gavioli, “On the benefits of elastic transponders in optical metro networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2012, paper OW3A.6.

Ramaswami, R.

O. Gerstel, R. Ramaswami, and G. Sasaki, “Cost-effective traffic grooming in WDM rings,” IEEE/ACM Trans. Netw., vol.  8, no. 5, pp. 618–630, Oct. 2000.
[CrossRef]

Rehman, M.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Rival, O.

O. Rival, A. Morea, and J.-C. Antona, “Optical network planning with rate-tunable NRZ transponders,” in Proc. ECOC, Vienna, Austria, Sept. 2009.

O. Rival and A. More, “Cost-efficiency of mixed 10-40-100  Gb/s networks and elastic optical networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI4.

Rottondi, C.

C. Rottondi, M. Tornatore, A. Pattavina, and G. Gavioli, “Traffic grooming and spectrum assignment for coherent transceivers in metro-flexible networks,” IEEE Photon. Technol. Lett., vol.  25, pp. 183–186, Jan. 2013.
[CrossRef]

C. Rottondi, M. Tornatore, F. Puleio, S. Raavi, A. Pattavina, and G. Gavioli, “On the benefits of elastic transponders in optical metro networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2012, paper OW3A.6.

Rouskas, G.

R. Dutta and G. Rouskas, “On optimal traffic grooming in WDM rings,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 110–121, Jan. 2002.
[CrossRef]

Sambo, N.

Sasaki, G.

O. Gerstel, R. Ramaswami, and G. Sasaki, “Cost-effective traffic grooming in WDM rings,” IEEE/ACM Trans. Netw., vol.  8, no. 5, pp. 618–630, Oct. 2000.
[CrossRef]

Schmidt, E. D.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Secondini, M.

Shen, G.

G. Shen and M. Zukerman, “Spectrum-efficient and agile CO-OFDM optical transport networks: architecture, design, and operation,” IEEE Commun. Mag., vol.  50, no. 5, pp. 82–89, May 2012.
[CrossRef]

G. Shen and Q. Yang, “From coarse grid to mini-grid to gridless: how much can gridless help contentionless?” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Simeonidou, D.

N. Amaya, I. Muhammad, G. S. Zervas, R. Nejabati, D. Simeonidou, Y. Zhou, and A. Lord, “Experimental demonstration of a gridless multi-granular optical network supporting flexible spectrum switching,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Sone, Y.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, Nov. 2009.
[CrossRef]

Spaelter, S.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Spinnler, B.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Stachowiak, J.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Sun, W.

W. Zheng, Y. Jin, W. Sun, W. Guo, and W. Hu, “On the spectrum-efficiency of bandwidth-variable optical OFDM transport networks,” in Optical Fiber Communication Conf., San Diego, CA, Mar. 2010, paper OWR5.

Takara, H.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, Nov. 2009.
[CrossRef]

Tanaka, T.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

Tomkos, I.

Tornatore, M.

C. Rottondi, M. Tornatore, A. Pattavina, and G. Gavioli, “Traffic grooming and spectrum assignment for coherent transceivers in metro-flexible networks,” IEEE Photon. Technol. Lett., vol.  25, pp. 183–186, Jan. 2013.
[CrossRef]

C. Rottondi, M. Tornatore, F. Puleio, S. Raavi, A. Pattavina, and G. Gavioli, “On the benefits of elastic transponders in optical metro networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2012, paper OW3A.6.

Tsukishima, Y.

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, Nov. 2009.
[CrossRef]

van den Borne, D.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Varvarigos, E. A.

Wang, C.

W. Wei, C. Wang, and X. Liu, “Adaptive IP/optical OFDM networking design,” in Optical Fiber Communication Conf., San Diego, CA, Mar. 2010, paper OWR6.

Wang, R.

R. Wang and B. Mukherjee, “Spectrum management in heterogeneous bandwidth networks,” in IEEE GLOBECOM, Anaheim, CA, Dec. 2012.

Wang, T.

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–204, July 2012.
[CrossRef]

W. Wei, J. Hu, D. Qian, P. Ji, T. Wang, X. Liu, and C. Qiao, “PONIARD: a programmable optical networking infrastructure for advanced research and development of future Internet,” J. Lightwave Technol., vol.  27, no. 3, pp. 233–242, 2009.
[CrossRef]

Wang, Y.

Y. Wang, X. Cao, and Y. Pan, “A study of the routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in Proc. IEEE INFOCOM, Shangai, China, 2011.

Y. Wang, X. Cao, and Q. Hu, “Routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in Proc. IEEE ICC, Kyoto, Japan, June 2011.

Watanabe, A.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

Wei, W.

Weiske, C. J.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Wellbrock, G.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Wuth, T.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Xia, T. J.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

Yang, Q.

G. Shen and Q. Yang, “From coarse grid to mini-grid to gridless: how much can gridless help contentionless?” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Zervas, G. S.

N. Amaya, I. Muhammad, G. S. Zervas, R. Nejabati, D. Simeonidou, Y. Zhou, and A. Lord, “Experimental demonstration of a gridless multi-granular optical network supporting flexible spectrum switching,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Zhang, H.

Y. Zhang, X. Zheng, Q. Li, N. Hua, Y. Li, and H. Zhang, “Traffic grooming in spectrum-elastic optical path networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI1.

Zhang, Y.

Y. Zhang, X. Zheng, Q. Li, N. Hua, Y. Li, and H. Zhang, “Traffic grooming in spectrum-elastic optical path networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI1.

Zheng, W.

W. Zheng, Y. Jin, W. Sun, W. Guo, and W. Hu, “On the spectrum-efficiency of bandwidth-variable optical OFDM transport networks,” in Optical Fiber Communication Conf., San Diego, CA, Mar. 2010, paper OWR5.

Zheng, X.

Y. Zhang, X. Zheng, Q. Li, N. Hua, Y. Li, and H. Zhang, “Traffic grooming in spectrum-elastic optical path networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI1.

Zhou, Y.

N. Amaya, I. Muhammad, G. S. Zervas, R. Nejabati, D. Simeonidou, Y. Zhou, and A. Lord, “Experimental demonstration of a gridless multi-granular optical network supporting flexible spectrum switching,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

Zukerman, M.

G. Shen and M. Zukerman, “Spectrum-efficient and agile CO-OFDM optical transport networks: architecture, design, and operation,” IEEE Commun. Mag., vol.  50, no. 5, pp. 82–89, May 2012.
[CrossRef]

IEEE Commun. Mag.

M. Jinno, B. Kozicki, H. Takara, A. Watanabe, Y. Sone, T. Tanaka, and A. Hirano, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, Aug. 2010.
[CrossRef]

G. Shen and M. Zukerman, “Spectrum-efficient and agile CO-OFDM optical transport networks: architecture, design, and operation,” IEEE Commun. Mag., vol.  50, no. 5, pp. 82–89, May 2012.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, Nov. 2009.
[CrossRef]

IEEE J. Sel. Areas Commun.

R. Dutta and G. Rouskas, “On optimal traffic grooming in WDM rings,” IEEE J. Sel. Areas Commun., vol.  20, no. 1, pp. 110–121, Jan. 2002.
[CrossRef]

IEEE Photon. Technol. Lett.

M. S. Alfiad, M. Kuschnerov, T. Wuth, T. J. Xia, G. Wellbrock, E. D. Schmidt, D. van den Borne, B. Spinnler, C. J. Weiske, E. de Man, A. Napoli, M. Finkenzeller, S. Spaelter, M. Rehman, J. Behel, M. Chbat, J. Stachowiak, D. Peterson, W. Lee, M. Pollock, B. Basch, D. Chen, M. Freiberger, B. Lankl, and H. de Waardt, “111  Gb/s transmission over 1040 km field-deployed fiber with 10  G/40  G neighbors,” IEEE Photon. Technol. Lett., vol.  21, no. 10, pp. 615–617, May 2009.
[CrossRef]

C. Rottondi, M. Tornatore, A. Pattavina, and G. Gavioli, “Traffic grooming and spectrum assignment for coherent transceivers in metro-flexible networks,” IEEE Photon. Technol. Lett., vol.  25, pp. 183–186, Jan. 2013.
[CrossRef]

IEEE/ACM Trans. Netw.

O. Gerstel, R. Ramaswami, and G. Sasaki, “Cost-effective traffic grooming in WDM rings,” IEEE/ACM Trans. Netw., vol.  8, no. 5, pp. 618–630, Oct. 2000.
[CrossRef]

J. Lightwave Technol.

Opt. Switching Networking

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–204, July 2012.
[CrossRef]

Other

Y. Wang, X. Cao, and Q. Hu, “Routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in Proc. IEEE ICC, Kyoto, Japan, June 2011.

Y. Wang, X. Cao, and Y. Pan, “A study of the routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in Proc. IEEE INFOCOM, Shangai, China, 2011.

W. Wei, C. Wang, and X. Liu, “Adaptive IP/optical OFDM networking design,” in Optical Fiber Communication Conf., San Diego, CA, Mar. 2010, paper OWR6.

C. Rottondi, M. Tornatore, F. Puleio, S. Raavi, A. Pattavina, and G. Gavioli, “On the benefits of elastic transponders in optical metro networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2012, paper OW3A.6.

G. Shen and Q. Yang, “From coarse grid to mini-grid to gridless: how much can gridless help contentionless?” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

ITU-T, Extension of Rec. G.694.1, Dec. 2011.

N. Amaya, I. Muhammad, G. S. Zervas, R. Nejabati, D. Simeonidou, Y. Zhou, and A. Lord, “Experimental demonstration of a gridless multi-granular optical network supporting flexible spectrum switching,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011.

W. Zheng, Y. Jin, W. Sun, W. Guo, and W. Hu, “On the spectrum-efficiency of bandwidth-variable optical OFDM transport networks,” in Optical Fiber Communication Conf., San Diego, CA, Mar. 2010, paper OWR5.

O. Rival, A. Morea, and J.-C. Antona, “Optical network planning with rate-tunable NRZ transponders,” in Proc. ECOC, Vienna, Austria, Sept. 2009.

B. Mukherjee, Optical WDM Networks.Springer, 2006.

R. Wang and B. Mukherjee, “Spectrum management in heterogeneous bandwidth networks,” in IEEE GLOBECOM, Anaheim, CA, Dec. 2012.

O. Rival and A. More, “Cost-efficiency of mixed 10-40-100  Gb/s networks and elastic optical networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI4.

Y. Zhang, X. Zheng, Q. Li, N. Hua, Y. Li, and H. Zhang, “Traffic grooming in spectrum-elastic optical path networks,” in Optical Fiber Communication Conf., Los Angeles, Mar. 2011, paper OTuI1.

S. Bucci, “Coherent technology: making 100  Gb/s Viable,” Techzine, Dec. 2011 [Online]. Available: http://www2.alcatel-lucent.com/blogs/techzine/2011/coherent-technology-making-100-gbs-viable/ .

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Figures (9)

Fig. 1.
Fig. 1.

Comparison of the architecture of a fixed and elastic transceiver.

Fig. 2.
Fig. 2.

Spectral occupancy of elastic superchannels.

Fig. 3.
Fig. 3.

Network architecture.

Fig. 4.
Fig. 4.

Spectrum occupation (in GHz) of elastic versus fixed WDM metro networks for all-to-all traffic matrices. The x axis indicates the traffic volume of a single traffic request.

Fig. 5.
Fig. 5.

Spectrum savings of elastic versus fixed WDM metro networks for all-to-all traffic matrices. The x axis indicates the traffic volume of a single traffic request.

Fig. 6.
Fig. 6.

Spectrum savings (in percentage) of elastic versus fixed WDM metro networks—dependence on the number of nodes (all-to-all traffic matrix, F = 6.25 GHZ , G = 6.25 GHZ , R = 100 km ).

Fig. 7.
Fig. 7.

Number of installed transceivers, considering all-to-all and one-to-all traffic matrices and different ring radius.

Fig. 8.
Fig. 8.

Average occupation of transceivers (all-to-all traffic matrix, R = 100 km , F = 6.25 GHz , G = 6.25 GHz ).

Fig. 9.
Fig. 9.

Fraction of lightpaths served by each modulation format (all-to-all traffic matrix, R = 100 km , F = 6.25 GHz , G = 6.25 GHz ).

Tables (6)

Tables Icon

TABLE I Analytical Bounds for the Spectrum Minimization Problem

Tables Icon

TABLE II Analytical Bounds for the Transceiver Minimization Problem

Tables Icon

Algorithm 1 Algorithm for the MinSpectrum Problem

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Algorithm 2 Algorithm for the MinTransceiver Problem

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TABLE III Reach Values of Various Modulation Formats (km) and for 28 GBd Fixed and Elastic Transceivers a

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TABLE IV Comparison of the Results Obtained With Algorithms 1 and 2 With Respect to Optimal Results

Equations (12)

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min α 1 i , j N , k K : i j 2 m i j k + α 2 i , j N , k K : i j L i j k ( b i j k F + y i j k G ) ,
k K x i j k s d 1 i , j , s , d N : i j , s d ,
s N , d N : s d x i j k s d t s d R i j k C b i j k i , j N , k K : i j ,
x i j k s d y i j k i , j , s , d N , k K : i j , s d ,
f i j k + b i j k F + G S i , j N , k K : i j ,
b i j k F m i j k T i , j N , k K : i j ,
k K , j N : i j x i j k s d k K , j N : i j x j i k s d = { 1 if s = i 1 if d = i 0 otherwise i , s , d N : s d t s d > 0 ,
F ( f i j k f i j k ) S d i j k i j k i , j , i , j N , k , k K : E k ( i j ) E k ( i j ) ,
F ( f i j k f i j k ) S d i j k i j k i , j , i , j N , k , k K : E k ( i j ) E k ( i j ) ,
d i j k , i j k + d i j k i j k = 1 i , j , i , j N , k , k K : E k ( i j ) E k ( i j ) ,
( f i j k + b i j k F ) + G f i j k F ( S + G ) ( 1 d i j k i j k + 2 y i j k y i j k ) i , j , i , j N , k , k K : E k ( i j ) E k ( i j ) ,
( f i j k + b i j k ) F + G f i j k F ( S + G ) ( 1 d i j k i j k + 2 y i j k y i j k ) i , j , i , j N , k , k K : E k ( i j ) E k ( i j )