Abstract

Using a graph-theoretic formulation, a grooming in a SONET ring network may be interpreted as a decomposition of an undirected simple graph G=(V,E), where V corresponds to the n nodes in the ring, and each edge {i,j}E represents the traffic requirements for the primitive ring {i,j}. In G={G1,,Gs}, the decomposition of G, each subgraph Gi specifies a set of primitive rings assigned to the same wavelength. If the maximum size set is c then G is a c-grooming. In this paper, bounding the maximum throughput tp¯(c,n,l) of a c-grooming G is addressed, subject to each node being equipped with a limited number l of add–drop multiplexers (ADMs). Naturally, restricting the number of ADMs limits the achievable throughput. For all l, precise determinations of maximum throughput for grooming ratios c=2, 3, and 4 are given. These underlie substantially improved bounds for larger grooming ratios.

© 2010 Optical Society of America

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References

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  1. E. Modiano, P. Lin, “Traffic grooming in WDM networks,” IEEE Commun. Mag., vol. 39, no. 7, pp. 124–129, 2001.
    [CrossRef]
  2. J.-C. Bermond, D. Coudert, “Traffic grooming in unidirectional WDM ring networks using design theory,” in IEEE Conf. on Communication (ICC’03), Los Alamitos, CA, 2003, vol. 2, pp. 1402–1406.
  3. J.-C. Bermond, D. Coudert, X. Muñoz, “Traffic grooming in unidirectional WDM ring networks: the all-to-all unitary case,” in ONDM’03: 7th IFIP Working Conf. on Optical Network Design and Modelling, 2003, pp. 1135–1153.
  4. R. Berry, E. Modiano, “Reducing electronic multiplexing costs in SONET/WDM rings with dynamically changing traffic,” IEEE J. Sel. Areas Commun., vol. 18, pp. 1961–1971, 2000.
    [CrossRef]
  5. R. Dutta, N. Rouskas, “Traffic grooming in WDM networks: past and future,” IEEE Network, vol. 16, no. 6, pp. 46–56, 2002.
    [CrossRef]
  6. P. J. Wan, Multichannel Optical Networks. Norwell, MA: Kluwer Academic, 2000.
  7. Y. Wang, Q.-P. Gu, “Maximum throughput traffic grooming in optical networks,” J. Opt. Netw., vol. 7, pp. 895–904, 2008.
    [CrossRef]
  8. O. Goldschmidt, D. Hochbaum, A. Levin, E. Olinick, “The SONET edge-partition problem,” Networks, vol. 41, pp. 13–23, 2003.
    [CrossRef]
  9. J. Hu, “Optimal traffic grooming for wavelength-division-multiplexing rings with all-to-all uniform traffic,” J. Opt. Netw., vol. 1, no. 1, pp. 32–42, 2002.
  10. C. J. Colbourn, G. Quattrocchi, V. R. Syrotiuk, “Grooming for two-period optical networks,” Networks, vol. 58, pp. 307–324, 2008.
    [CrossRef]
  11. P. Adams, D. Bryant, M. Buchanan, “A survey on the existence of G-designs,” J. Comb. Designs, vol. 16, pp. 373–410, 2008.
    [CrossRef]
  12. J.-C. Bermond, S. Ceroi, “Minimizing SONET ADMs in unidirectional WDM rings with grooming ratio 3,” Networks, vol. 41, pp. 83–86, 2003.
    [CrossRef]
  13. J.-C. Bermond, C. J. Colbourn, A. C. H. Ling, M. L. Yu, “Grooming in unidirectional rings: K4−e designs,” Discrete Math., vol. 284, pp. 67–72, 2004.
    [CrossRef]
  14. C. J. Colbourn, G. Ge, A. C. H. Ling, “Optimal grooming with grooming ratio nine,” Discrete Math., to be published.
  15. J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, I. Sau, “Drop cost and wavelength optimal two-period grooming with ratio 4,” SIAM J. Discrete Mathematics, vol. 24, pp. 400–419, 2010.
    [CrossRef]
  16. K. Zhu, B. Mukherjee, “Traffic grooming in an optical WDM mesh network,” IEEE J. Sel. Areas Commun., vol. 20, no. 1, pp. 122–133, 2002.
    [CrossRef]
  17. N. Srinivas, C. S. R. Murthy, “Throughput maximization in traffic grooming in WDM mesh networks,” J. High Speed Networks, vol. 13, pp. 139–154, 2004.
  18. W. Yao, B. Ramamurthy, “Rerouting schemes for dynamic traffic grooming in optical WDM networks,” Comput. Netw., vol. 52, pp. 1891–1904, 2008.
    [CrossRef]
  19. N. S. C. Correia, M. C. R. Medeiros, “Traffic grooming applied to network protection: throughput and grooming port cost analysis,” in Proc. of the 7th Int. Conf. on Transparent Optical Networks (ICTON’05), 2005, pp. 389–393.
  20. P. Prathombutr, J. Stach, E. K. Park, “An algorithm for traffic grooming in WDM optical mesh networks with multiple objectives,” Telecommun. Syst., vol. 28, no. 3,4, pp. 369–386, 2005.
    [CrossRef]
  21. P. Dukes, A. C. H. Ling, “Asymptotic existence of resolvable graph designs,” Can. Math. Bull., vol. 50, no. 4, pp. 504–518, 2007.
    [CrossRef]
  22. D. de Werra, “Equitable colorations of graphs,” Rev. Fr. Inform. Rech. Oper., vol. 5, no. 3, pp. 3–8, 1971.
  23. E. B. Yavorskii, “Representations of directed graphs and ψ-transformations,” in Theoretical and Applied Questions of Differential Equations and Algebra, A. N. Sharkovskii, Ed. Naukova Dumka, 1978, pp. 247–250.
  24. C. J. Colbourn, A. Rosa, Triple Systems. Oxford: Oxford U. Press, 1999.
  25. L. D. Andersen, A. J. W. Hilton, E. Mendelsohn, “Embedding partial Steiner triple systems,” Proc. London Math. Soc., vol. 41, no. 3, pp. 557–576, 1980.
    [CrossRef]

2010

J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, I. Sau, “Drop cost and wavelength optimal two-period grooming with ratio 4,” SIAM J. Discrete Mathematics, vol. 24, pp. 400–419, 2010.
[CrossRef]

2008

C. J. Colbourn, G. Quattrocchi, V. R. Syrotiuk, “Grooming for two-period optical networks,” Networks, vol. 58, pp. 307–324, 2008.
[CrossRef]

P. Adams, D. Bryant, M. Buchanan, “A survey on the existence of G-designs,” J. Comb. Designs, vol. 16, pp. 373–410, 2008.
[CrossRef]

W. Yao, B. Ramamurthy, “Rerouting schemes for dynamic traffic grooming in optical WDM networks,” Comput. Netw., vol. 52, pp. 1891–1904, 2008.
[CrossRef]

Y. Wang, Q.-P. Gu, “Maximum throughput traffic grooming in optical networks,” J. Opt. Netw., vol. 7, pp. 895–904, 2008.
[CrossRef]

2007

P. Dukes, A. C. H. Ling, “Asymptotic existence of resolvable graph designs,” Can. Math. Bull., vol. 50, no. 4, pp. 504–518, 2007.
[CrossRef]

2005

P. Prathombutr, J. Stach, E. K. Park, “An algorithm for traffic grooming in WDM optical mesh networks with multiple objectives,” Telecommun. Syst., vol. 28, no. 3,4, pp. 369–386, 2005.
[CrossRef]

2004

J.-C. Bermond, C. J. Colbourn, A. C. H. Ling, M. L. Yu, “Grooming in unidirectional rings: K4−e designs,” Discrete Math., vol. 284, pp. 67–72, 2004.
[CrossRef]

N. Srinivas, C. S. R. Murthy, “Throughput maximization in traffic grooming in WDM mesh networks,” J. High Speed Networks, vol. 13, pp. 139–154, 2004.

2003

J.-C. Bermond, S. Ceroi, “Minimizing SONET ADMs in unidirectional WDM rings with grooming ratio 3,” Networks, vol. 41, pp. 83–86, 2003.
[CrossRef]

O. Goldschmidt, D. Hochbaum, A. Levin, E. Olinick, “The SONET edge-partition problem,” Networks, vol. 41, pp. 13–23, 2003.
[CrossRef]

2002

J. Hu, “Optimal traffic grooming for wavelength-division-multiplexing rings with all-to-all uniform traffic,” J. Opt. Netw., vol. 1, no. 1, pp. 32–42, 2002.

R. Dutta, N. Rouskas, “Traffic grooming in WDM networks: past and future,” IEEE Network, vol. 16, no. 6, pp. 46–56, 2002.
[CrossRef]

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

2001

E. Modiano, P. Lin, “Traffic grooming in WDM networks,” IEEE Commun. Mag., vol. 39, no. 7, pp. 124–129, 2001.
[CrossRef]

2000

R. Berry, E. Modiano, “Reducing electronic multiplexing costs in SONET/WDM rings with dynamically changing traffic,” IEEE J. Sel. Areas Commun., vol. 18, pp. 1961–1971, 2000.
[CrossRef]

1980

L. D. Andersen, A. J. W. Hilton, E. Mendelsohn, “Embedding partial Steiner triple systems,” Proc. London Math. Soc., vol. 41, no. 3, pp. 557–576, 1980.
[CrossRef]

1971

D. de Werra, “Equitable colorations of graphs,” Rev. Fr. Inform. Rech. Oper., vol. 5, no. 3, pp. 3–8, 1971.

Adams, P.

P. Adams, D. Bryant, M. Buchanan, “A survey on the existence of G-designs,” J. Comb. Designs, vol. 16, pp. 373–410, 2008.
[CrossRef]

Andersen, L. D.

L. D. Andersen, A. J. W. Hilton, E. Mendelsohn, “Embedding partial Steiner triple systems,” Proc. London Math. Soc., vol. 41, no. 3, pp. 557–576, 1980.
[CrossRef]

Bermond, J.-C.

J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, I. Sau, “Drop cost and wavelength optimal two-period grooming with ratio 4,” SIAM J. Discrete Mathematics, vol. 24, pp. 400–419, 2010.
[CrossRef]

J.-C. Bermond, C. J. Colbourn, A. C. H. Ling, M. L. Yu, “Grooming in unidirectional rings: K4−e designs,” Discrete Math., vol. 284, pp. 67–72, 2004.
[CrossRef]

J.-C. Bermond, S. Ceroi, “Minimizing SONET ADMs in unidirectional WDM rings with grooming ratio 3,” Networks, vol. 41, pp. 83–86, 2003.
[CrossRef]

J.-C. Bermond, D. Coudert, “Traffic grooming in unidirectional WDM ring networks using design theory,” in IEEE Conf. on Communication (ICC’03), Los Alamitos, CA, 2003, vol. 2, pp. 1402–1406.

J.-C. Bermond, D. Coudert, X. Muñoz, “Traffic grooming in unidirectional WDM ring networks: the all-to-all unitary case,” in ONDM’03: 7th IFIP Working Conf. on Optical Network Design and Modelling, 2003, pp. 1135–1153.

Berry, R.

R. Berry, E. Modiano, “Reducing electronic multiplexing costs in SONET/WDM rings with dynamically changing traffic,” IEEE J. Sel. Areas Commun., vol. 18, pp. 1961–1971, 2000.
[CrossRef]

Bryant, D.

P. Adams, D. Bryant, M. Buchanan, “A survey on the existence of G-designs,” J. Comb. Designs, vol. 16, pp. 373–410, 2008.
[CrossRef]

Buchanan, M.

P. Adams, D. Bryant, M. Buchanan, “A survey on the existence of G-designs,” J. Comb. Designs, vol. 16, pp. 373–410, 2008.
[CrossRef]

Ceroi, S.

J.-C. Bermond, S. Ceroi, “Minimizing SONET ADMs in unidirectional WDM rings with grooming ratio 3,” Networks, vol. 41, pp. 83–86, 2003.
[CrossRef]

Colbourn, C. J.

J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, I. Sau, “Drop cost and wavelength optimal two-period grooming with ratio 4,” SIAM J. Discrete Mathematics, vol. 24, pp. 400–419, 2010.
[CrossRef]

C. J. Colbourn, G. Quattrocchi, V. R. Syrotiuk, “Grooming for two-period optical networks,” Networks, vol. 58, pp. 307–324, 2008.
[CrossRef]

J.-C. Bermond, C. J. Colbourn, A. C. H. Ling, M. L. Yu, “Grooming in unidirectional rings: K4−e designs,” Discrete Math., vol. 284, pp. 67–72, 2004.
[CrossRef]

C. J. Colbourn, A. Rosa, Triple Systems. Oxford: Oxford U. Press, 1999.

C. J. Colbourn, G. Ge, A. C. H. Ling, “Optimal grooming with grooming ratio nine,” Discrete Math., to be published.

Correia, N. S. C.

N. S. C. Correia, M. C. R. Medeiros, “Traffic grooming applied to network protection: throughput and grooming port cost analysis,” in Proc. of the 7th Int. Conf. on Transparent Optical Networks (ICTON’05), 2005, pp. 389–393.

Coudert, D.

J.-C. Bermond, D. Coudert, X. Muñoz, “Traffic grooming in unidirectional WDM ring networks: the all-to-all unitary case,” in ONDM’03: 7th IFIP Working Conf. on Optical Network Design and Modelling, 2003, pp. 1135–1153.

J.-C. Bermond, D. Coudert, “Traffic grooming in unidirectional WDM ring networks using design theory,” in IEEE Conf. on Communication (ICC’03), Los Alamitos, CA, 2003, vol. 2, pp. 1402–1406.

de Werra, D.

D. de Werra, “Equitable colorations of graphs,” Rev. Fr. Inform. Rech. Oper., vol. 5, no. 3, pp. 3–8, 1971.

Dukes, P.

P. Dukes, A. C. H. Ling, “Asymptotic existence of resolvable graph designs,” Can. Math. Bull., vol. 50, no. 4, pp. 504–518, 2007.
[CrossRef]

Dutta, R.

R. Dutta, N. Rouskas, “Traffic grooming in WDM networks: past and future,” IEEE Network, vol. 16, no. 6, pp. 46–56, 2002.
[CrossRef]

Ge, G.

C. J. Colbourn, G. Ge, A. C. H. Ling, “Optimal grooming with grooming ratio nine,” Discrete Math., to be published.

Gionfriddo, L.

J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, I. Sau, “Drop cost and wavelength optimal two-period grooming with ratio 4,” SIAM J. Discrete Mathematics, vol. 24, pp. 400–419, 2010.
[CrossRef]

Goldschmidt, O.

O. Goldschmidt, D. Hochbaum, A. Levin, E. Olinick, “The SONET edge-partition problem,” Networks, vol. 41, pp. 13–23, 2003.
[CrossRef]

Gu, Q.-P.

Hilton, A. J. W.

L. D. Andersen, A. J. W. Hilton, E. Mendelsohn, “Embedding partial Steiner triple systems,” Proc. London Math. Soc., vol. 41, no. 3, pp. 557–576, 1980.
[CrossRef]

Hochbaum, D.

O. Goldschmidt, D. Hochbaum, A. Levin, E. Olinick, “The SONET edge-partition problem,” Networks, vol. 41, pp. 13–23, 2003.
[CrossRef]

Hu, J.

Levin, A.

O. Goldschmidt, D. Hochbaum, A. Levin, E. Olinick, “The SONET edge-partition problem,” Networks, vol. 41, pp. 13–23, 2003.
[CrossRef]

Lin, P.

E. Modiano, P. Lin, “Traffic grooming in WDM networks,” IEEE Commun. Mag., vol. 39, no. 7, pp. 124–129, 2001.
[CrossRef]

Ling, A. C. H.

P. Dukes, A. C. H. Ling, “Asymptotic existence of resolvable graph designs,” Can. Math. Bull., vol. 50, no. 4, pp. 504–518, 2007.
[CrossRef]

J.-C. Bermond, C. J. Colbourn, A. C. H. Ling, M. L. Yu, “Grooming in unidirectional rings: K4−e designs,” Discrete Math., vol. 284, pp. 67–72, 2004.
[CrossRef]

C. J. Colbourn, G. Ge, A. C. H. Ling, “Optimal grooming with grooming ratio nine,” Discrete Math., to be published.

Medeiros, M. C. R.

N. S. C. Correia, M. C. R. Medeiros, “Traffic grooming applied to network protection: throughput and grooming port cost analysis,” in Proc. of the 7th Int. Conf. on Transparent Optical Networks (ICTON’05), 2005, pp. 389–393.

Mendelsohn, E.

L. D. Andersen, A. J. W. Hilton, E. Mendelsohn, “Embedding partial Steiner triple systems,” Proc. London Math. Soc., vol. 41, no. 3, pp. 557–576, 1980.
[CrossRef]

Modiano, E.

E. Modiano, P. Lin, “Traffic grooming in WDM networks,” IEEE Commun. Mag., vol. 39, no. 7, pp. 124–129, 2001.
[CrossRef]

R. Berry, E. Modiano, “Reducing electronic multiplexing costs in SONET/WDM rings with dynamically changing traffic,” IEEE J. Sel. Areas Commun., vol. 18, pp. 1961–1971, 2000.
[CrossRef]

Mukherjee, B.

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

Muñoz, X.

J.-C. Bermond, D. Coudert, X. Muñoz, “Traffic grooming in unidirectional WDM ring networks: the all-to-all unitary case,” in ONDM’03: 7th IFIP Working Conf. on Optical Network Design and Modelling, 2003, pp. 1135–1153.

Murthy, C. S. R.

N. Srinivas, C. S. R. Murthy, “Throughput maximization in traffic grooming in WDM mesh networks,” J. High Speed Networks, vol. 13, pp. 139–154, 2004.

Olinick, E.

O. Goldschmidt, D. Hochbaum, A. Levin, E. Olinick, “The SONET edge-partition problem,” Networks, vol. 41, pp. 13–23, 2003.
[CrossRef]

Park, E. K.

P. Prathombutr, J. Stach, E. K. Park, “An algorithm for traffic grooming in WDM optical mesh networks with multiple objectives,” Telecommun. Syst., vol. 28, no. 3,4, pp. 369–386, 2005.
[CrossRef]

Prathombutr, P.

P. Prathombutr, J. Stach, E. K. Park, “An algorithm for traffic grooming in WDM optical mesh networks with multiple objectives,” Telecommun. Syst., vol. 28, no. 3,4, pp. 369–386, 2005.
[CrossRef]

Quattrocchi, G.

J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, I. Sau, “Drop cost and wavelength optimal two-period grooming with ratio 4,” SIAM J. Discrete Mathematics, vol. 24, pp. 400–419, 2010.
[CrossRef]

C. J. Colbourn, G. Quattrocchi, V. R. Syrotiuk, “Grooming for two-period optical networks,” Networks, vol. 58, pp. 307–324, 2008.
[CrossRef]

Ramamurthy, B.

W. Yao, B. Ramamurthy, “Rerouting schemes for dynamic traffic grooming in optical WDM networks,” Comput. Netw., vol. 52, pp. 1891–1904, 2008.
[CrossRef]

Rosa, A.

C. J. Colbourn, A. Rosa, Triple Systems. Oxford: Oxford U. Press, 1999.

Rouskas, N.

R. Dutta, N. Rouskas, “Traffic grooming in WDM networks: past and future,” IEEE Network, vol. 16, no. 6, pp. 46–56, 2002.
[CrossRef]

Sau, I.

J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, I. Sau, “Drop cost and wavelength optimal two-period grooming with ratio 4,” SIAM J. Discrete Mathematics, vol. 24, pp. 400–419, 2010.
[CrossRef]

Srinivas, N.

N. Srinivas, C. S. R. Murthy, “Throughput maximization in traffic grooming in WDM mesh networks,” J. High Speed Networks, vol. 13, pp. 139–154, 2004.

Stach, J.

P. Prathombutr, J. Stach, E. K. Park, “An algorithm for traffic grooming in WDM optical mesh networks with multiple objectives,” Telecommun. Syst., vol. 28, no. 3,4, pp. 369–386, 2005.
[CrossRef]

Syrotiuk, V. R.

C. J. Colbourn, G. Quattrocchi, V. R. Syrotiuk, “Grooming for two-period optical networks,” Networks, vol. 58, pp. 307–324, 2008.
[CrossRef]

Wan, P. J.

P. J. Wan, Multichannel Optical Networks. Norwell, MA: Kluwer Academic, 2000.

Wang, Y.

Yao, W.

W. Yao, B. Ramamurthy, “Rerouting schemes for dynamic traffic grooming in optical WDM networks,” Comput. Netw., vol. 52, pp. 1891–1904, 2008.
[CrossRef]

Yavorskii, E. B.

E. B. Yavorskii, “Representations of directed graphs and ψ-transformations,” in Theoretical and Applied Questions of Differential Equations and Algebra, A. N. Sharkovskii, Ed. Naukova Dumka, 1978, pp. 247–250.

Yu, M. L.

J.-C. Bermond, C. J. Colbourn, A. C. H. Ling, M. L. Yu, “Grooming in unidirectional rings: K4−e designs,” Discrete Math., vol. 284, pp. 67–72, 2004.
[CrossRef]

Zhu, K.

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

Can. Math. Bull.

P. Dukes, A. C. H. Ling, “Asymptotic existence of resolvable graph designs,” Can. Math. Bull., vol. 50, no. 4, pp. 504–518, 2007.
[CrossRef]

Comput. Netw.

W. Yao, B. Ramamurthy, “Rerouting schemes for dynamic traffic grooming in optical WDM networks,” Comput. Netw., vol. 52, pp. 1891–1904, 2008.
[CrossRef]

Discrete Math.

J.-C. Bermond, C. J. Colbourn, A. C. H. Ling, M. L. Yu, “Grooming in unidirectional rings: K4−e designs,” Discrete Math., vol. 284, pp. 67–72, 2004.
[CrossRef]

C. J. Colbourn, G. Ge, A. C. H. Ling, “Optimal grooming with grooming ratio nine,” Discrete Math., to be published.

IEEE Commun. Mag.

E. Modiano, P. Lin, “Traffic grooming in WDM networks,” IEEE Commun. Mag., vol. 39, no. 7, pp. 124–129, 2001.
[CrossRef]

IEEE J. Sel. Areas Commun.

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

R. Berry, E. Modiano, “Reducing electronic multiplexing costs in SONET/WDM rings with dynamically changing traffic,” IEEE J. Sel. Areas Commun., vol. 18, pp. 1961–1971, 2000.
[CrossRef]

IEEE Network

R. Dutta, N. Rouskas, “Traffic grooming in WDM networks: past and future,” IEEE Network, vol. 16, no. 6, pp. 46–56, 2002.
[CrossRef]

J. Comb. Designs

P. Adams, D. Bryant, M. Buchanan, “A survey on the existence of G-designs,” J. Comb. Designs, vol. 16, pp. 373–410, 2008.
[CrossRef]

J. High Speed Networks

N. Srinivas, C. S. R. Murthy, “Throughput maximization in traffic grooming in WDM mesh networks,” J. High Speed Networks, vol. 13, pp. 139–154, 2004.

J. Opt. Netw.

Networks

O. Goldschmidt, D. Hochbaum, A. Levin, E. Olinick, “The SONET edge-partition problem,” Networks, vol. 41, pp. 13–23, 2003.
[CrossRef]

J.-C. Bermond, S. Ceroi, “Minimizing SONET ADMs in unidirectional WDM rings with grooming ratio 3,” Networks, vol. 41, pp. 83–86, 2003.
[CrossRef]

C. J. Colbourn, G. Quattrocchi, V. R. Syrotiuk, “Grooming for two-period optical networks,” Networks, vol. 58, pp. 307–324, 2008.
[CrossRef]

Proc. London Math. Soc.

L. D. Andersen, A. J. W. Hilton, E. Mendelsohn, “Embedding partial Steiner triple systems,” Proc. London Math. Soc., vol. 41, no. 3, pp. 557–576, 1980.
[CrossRef]

Rev. Fr. Inform. Rech. Oper.

D. de Werra, “Equitable colorations of graphs,” Rev. Fr. Inform. Rech. Oper., vol. 5, no. 3, pp. 3–8, 1971.

SIAM J. Discrete Mathematics

J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, I. Sau, “Drop cost and wavelength optimal two-period grooming with ratio 4,” SIAM J. Discrete Mathematics, vol. 24, pp. 400–419, 2010.
[CrossRef]

Telecommun. Syst.

P. Prathombutr, J. Stach, E. K. Park, “An algorithm for traffic grooming in WDM optical mesh networks with multiple objectives,” Telecommun. Syst., vol. 28, no. 3,4, pp. 369–386, 2005.
[CrossRef]

Other

J.-C. Bermond, D. Coudert, “Traffic grooming in unidirectional WDM ring networks using design theory,” in IEEE Conf. on Communication (ICC’03), Los Alamitos, CA, 2003, vol. 2, pp. 1402–1406.

J.-C. Bermond, D. Coudert, X. Muñoz, “Traffic grooming in unidirectional WDM ring networks: the all-to-all unitary case,” in ONDM’03: 7th IFIP Working Conf. on Optical Network Design and Modelling, 2003, pp. 1135–1153.

E. B. Yavorskii, “Representations of directed graphs and ψ-transformations,” in Theoretical and Applied Questions of Differential Equations and Algebra, A. N. Sharkovskii, Ed. Naukova Dumka, 1978, pp. 247–250.

C. J. Colbourn, A. Rosa, Triple Systems. Oxford: Oxford U. Press, 1999.

N. S. C. Correia, M. C. R. Medeiros, “Traffic grooming applied to network protection: throughput and grooming port cost analysis,” in Proc. of the 7th Int. Conf. on Transparent Optical Networks (ICTON’05), 2005, pp. 389–393.

P. J. Wan, Multichannel Optical Networks. Norwell, MA: Kluwer Academic, 2000.

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Figures (2)

Fig. 1
Fig. 1

A graph decomposition G = { G 1 , G 2 , G 3 } of the complete graph K 4 .

Fig. 2
Fig. 2

A SONET ring network realizing the parameters of the graph decomposition in Fig. 1.

Equations (6)

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

d c ̱ ( 3 , n , ( n 2 ) ) = { ( n 2 ) if n 1 , 3 ( mod 6 ) ( n 2 ) + 2 if n 5 ( mod 6 ) ( n 2 ) + n 4 if n 0 , 2 , 4 , 6 , 10 ( mod 12 ) ( n 2 ) + n 4 + 1 if n 8 ( mod 12 ) . }
d c ̱ ( 5 , n , ( n 2 ) ) = 4 1 5 ( n 2 ) + { 0 if n 0 , 1 ( mod 5 ) and n 5 1 if n = 5 2 if n 2 , 4 ( mod 5 ) and n 7 3 if n = 7 3 if n 3 ( mod 5 ) and n 8 4 if n = 8. }
t p ¯ ( c , n , l ) ( c ) 2 l n c 2 n l c 2 .
t p ¯ ( c , n , l ) ( c c ) 2 ( ( x 1 ) n c c l 2 κ c ) n l c x 1 2 x ,
t p ¯ ( 3 , n , l ) = { n l if l n 1 2 and n l 0 ( mod 3 ) ( n 1 ) l if l n 1 2 and n l 0 ( mod 3 ) ( n 2 ) otherwise . }
m n = { n ( n 2 ) 2 if n 0 , 2 ( mod 6 ) ( n 2 ) if n 1 , 3 ( mod 6 ) n ( n 2 ) 2 1 if n 4 ( mod 6 ) ( n 2 ) 4 if n 5 ( mod 6 ) . }