Abstract

In this work, we establish analytic bounds for the energy efficiency of 1+1 survivable IP over WDM networks using network coding. The analytic bounds are shown to be in close agreement with our previously reported results. They provide verification of the MILP and heuristics proposed previously, in addition to an efficient, compact means to evaluate network results and allow the performance of large networks to be determined easily.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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  1. R. Ahlswede, C. Ning, S. Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol.  46, no. 4, pp. 1204–1216, 2000.
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  2. A. E. Kamal and M. Mohandespour, “Network coding-based protection,” Opt. Switching Netw., vol.  11, pp. 189–201, 2014.
    [Crossref]
  3. P. Babarczi, G. Biczok, H. Øverby, J. Tapolcai, and P. Soproni, “Realization strategies of dedicated path protection: a bandwidth cost perspective,” Comput. Netw., vol.  57, no. 9, pp. 1974–1990, 2013.
    [Crossref]
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    [Crossref]
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  19. A. Lawey, T. El-Gorashi, and J. Elmirghani, “Distributed energy efficient clouds over core networks,” J. Lightwave Technol., vol.  32, no. 7, pp. 1261–1281, Apr.2014.
    [Crossref]
  20. N. Osman, T. El-Gorashi, L. Krug, and J. Elmirghani, “Energy-efficient future high-definition TV,” J. Lightwave Technol., vol.  32, no. 13, pp. 2364–2381, July2014.
    [Crossref]
  21. A. Lawey, T. El-Gorashi, and J. Elmirghani, “BitTorrent content distribution in optical networks,” J. Lightwave Technol., vol.  32, no. 21, pp. 4209–4225, Nov.2014.
    [Crossref]
  22. L. Nonde, T. El-Gorashi, and J. Elmirghani, “Energy efficient virtual network embedding for cloud networks,” J. Lightwave Technol., vol.  33, no. 9, pp. 1828–1849, May2015.
    [Crossref]
  23. M. Musa, T. El-Gorashi, and J. Elmirghani, “Energy efficient routing and network coding assignment in core networks,” J. Lightwave Technol. (to be published).
  24. M. Musa, T. Elgorashi, and J. Elmirghani, “Energy efficient survivable IP-over-WDM networks with network coding,” J. Opt. Commun. Netw., vol.  9, no. 3, pp. 207–217, 2017.
    [Crossref]
  25. A. S. Thyagaturu, A. Mercian, M. P. McGarry, M. Reisslein, and W. Kellerer, “Software defined optical networks (SDONs): a comprehensive survey,” Commun. Surv. Tutorials, vol.  18, no. 4, pp. 2738–2786, 2016.
    [Crossref]
  26. A. Capone, C. Cascone, A. Q. T. Nguyen, and B. Sanso, “Detour planning for fast and reliable failure recovery in SDN with OpenState,” in 11th Int. Conf. on the Design of Reliable Communication Networks (DRCN), IEEE, 2015.

2017 (1)

2016 (2)

A. S. Thyagaturu, A. Mercian, M. P. McGarry, M. Reisslein, and W. Kellerer, “Software defined optical networks (SDONs): a comprehensive survey,” Commun. Surv. Tutorials, vol.  18, no. 4, pp. 2738–2786, 2016.
[Crossref]

F. Idzikowski, L. Chiaraviglio, A. Cianfrani, J. L. Vizcano, M. Polverini, and Y. Ye, “A survey on energy-aware design and operation of core networks,” Commun. Surveys Tuts., vol.  18, no. 2, pp. 1453–1499, 2016.
[Crossref]

2015 (2)

M. N. Dharmaweera, R. Parthiban, and Y. A. Sekercioglu, “Toward a power-efficient backbone network: the state of research,” Commun. Surv. Tutorials, vol.  17, no. 1, pp. 198–227, 2015.
[Crossref]

L. Nonde, T. El-Gorashi, and J. Elmirghani, “Energy efficient virtual network embedding for cloud networks,” J. Lightwave Technol., vol.  33, no. 9, pp. 1828–1849, May2015.
[Crossref]

2014 (4)

A. Lawey, T. El-Gorashi, and J. Elmirghani, “Distributed energy efficient clouds over core networks,” J. Lightwave Technol., vol.  32, no. 7, pp. 1261–1281, Apr.2014.
[Crossref]

N. Osman, T. El-Gorashi, L. Krug, and J. Elmirghani, “Energy-efficient future high-definition TV,” J. Lightwave Technol., vol.  32, no. 13, pp. 2364–2381, July2014.
[Crossref]

A. Lawey, T. El-Gorashi, and J. Elmirghani, “BitTorrent content distribution in optical networks,” J. Lightwave Technol., vol.  32, no. 21, pp. 4209–4225, Nov.2014.
[Crossref]

A. E. Kamal and M. Mohandespour, “Network coding-based protection,” Opt. Switching Netw., vol.  11, pp. 189–201, 2014.
[Crossref]

2013 (1)

P. Babarczi, G. Biczok, H. Øverby, J. Tapolcai, and P. Soproni, “Realization strategies of dedicated path protection: a bandwidth cost perspective,” Comput. Netw., vol.  57, no. 9, pp. 1974–1990, 2013.
[Crossref]

2012 (1)

2011 (3)

2010 (1)

A. E. Kamal, “Network protection for mesh networks: network coding based protection using p-cycles,” IEEE/ACM Trans. Netw., vol.  18, no. 1, pp. 67–80, 2010.
[Crossref]

2000 (1)

R. Ahlswede, C. Ning, S. Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol.  46, no. 4, pp. 1204–1216, 2000.
[Crossref]

Ahlswede, R.

R. Ahlswede, C. Ning, S. Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol.  46, no. 4, pp. 1204–1216, 2000.
[Crossref]

Al-Kofahi, O.

A. E. Kamal and O. Al-Kofahi, “Efficient and agile 1 + n protection,” IEEE Trans. Commun., vol.  59, no. 1, pp. 169–180, 2011.
[Crossref]

Aly, S. A.

S. A. Aly and A. E. Kamal, “Network protection codes against link failures using network coding,” in IEEE Global Telecommunications Conf., IEEE, 2008, pp. 1–6.

S. A. Aly and A. E. Kamal, “Network coding-based protection strategy against node failures,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2009, pp. 1–5.

Babarczi, P.

P. Babarczi, G. Biczok, H. Øverby, J. Tapolcai, and P. Soproni, “Realization strategies of dedicated path protection: a bandwidth cost perspective,” Comput. Netw., vol.  57, no. 9, pp. 1974–1990, 2013.
[Crossref]

H. Overby, G. Biczok, P. Babarczi, and J. Tapolcai, “Cost comparison of 1 + 1 path protection schemes: a case for coding,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2012.

Barla, I. B.

I. B. Barla, F. Rambach, D. A. Schupke, and M. Thakur, “Network coding for protection against multiple link failures in multi-domain networks,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2010, pp. 1–6

Biczok, G.

P. Babarczi, G. Biczok, H. Øverby, J. Tapolcai, and P. Soproni, “Realization strategies of dedicated path protection: a bandwidth cost perspective,” Comput. Netw., vol.  57, no. 9, pp. 1974–1990, 2013.
[Crossref]

H. Overby, G. Biczok, P. Babarczi, and J. Tapolcai, “Cost comparison of 1 + 1 path protection schemes: a case for coding,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2012.

Capone, A.

A. Capone, C. Cascone, A. Q. T. Nguyen, and B. Sanso, “Detour planning for fast and reliable failure recovery in SDN with OpenState,” in 11th Int. Conf. on the Design of Reliable Communication Networks (DRCN), IEEE, 2015.

Cascone, C.

A. Capone, C. Cascone, A. Q. T. Nguyen, and B. Sanso, “Detour planning for fast and reliable failure recovery in SDN with OpenState,” in 11th Int. Conf. on the Design of Reliable Communication Networks (DRCN), IEEE, 2015.

Chiaraviglio, L.

F. Idzikowski, L. Chiaraviglio, A. Cianfrani, J. L. Vizcano, M. Polverini, and Y. Ye, “A survey on energy-aware design and operation of core networks,” Commun. Surveys Tuts., vol.  18, no. 2, pp. 1453–1499, 2016.
[Crossref]

Cianfrani, A.

F. Idzikowski, L. Chiaraviglio, A. Cianfrani, J. L. Vizcano, M. Polverini, and Y. Ye, “A survey on energy-aware design and operation of core networks,” Commun. Surveys Tuts., vol.  18, no. 2, pp. 1453–1499, 2016.
[Crossref]

Dharmaweera, M. N.

M. N. Dharmaweera, R. Parthiban, and Y. A. Sekercioglu, “Toward a power-efficient backbone network: the state of research,” Commun. Surv. Tutorials, vol.  17, no. 1, pp. 198–227, 2015.
[Crossref]

Dong, X.

Elgorashi, T.

El-Gorashi, T.

El-Gorashi, T. E.

M. Musa, T. E. El-Gorashi, and J. M. Elmirghani, “Energy efficient core networks using network coding,” in 17th Int. Conf. on Transparent Optical Networks (ICTON), IEEE, 2015, p. 14.

M. Musa, T. E. El-Gorashi, and J. M. Elmirghani, “Network coding for energy efficiency in bypass IP/WDM networks,” in 18th Int. Conf. on Transparent Optical Networks (ICTON), IEEE, 2016.

El-Gorashi, T. E. H.

Elmirghani, J.

Elmirghani, J. M.

X. Dong, T. El-Gorashi, and J. M. Elmirghani, “IP over WDM networks employing renewable energy sources,” J. Lightwave Technol., vol.  29, no. 1, pp. 3–14, 2011.
[Crossref]

M. Musa, T. E. El-Gorashi, and J. M. Elmirghani, “Network coding for energy efficiency in bypass IP/WDM networks,” in 18th Int. Conf. on Transparent Optical Networks (ICTON), IEEE, 2016.

M. Musa, T. E. El-Gorashi, and J. M. Elmirghani, “Energy efficient core networks using network coding,” in 17th Int. Conf. on Transparent Optical Networks (ICTON), IEEE, 2015, p. 14.

Elmirghani, J. M. H.

Idzikowski, F.

F. Idzikowski, L. Chiaraviglio, A. Cianfrani, J. L. Vizcano, M. Polverini, and Y. Ye, “A survey on energy-aware design and operation of core networks,” Commun. Surveys Tuts., vol.  18, no. 2, pp. 1453–1499, 2016.
[Crossref]

Jose, A. A.

A. Muktadir, A. A. Jose, and E. Oki, “An optimum mathematical programming model for network-coding based routing with 1 + 1 path protection,” in World Telecommunications Congress (WTC), IEEE, 2012, pp. 1–5.

Kamal, A. E.

A. E. Kamal and M. Mohandespour, “Network coding-based protection,” Opt. Switching Netw., vol.  11, pp. 189–201, 2014.
[Crossref]

A. E. Kamal and O. Al-Kofahi, “Efficient and agile 1 + n protection,” IEEE Trans. Commun., vol.  59, no. 1, pp. 169–180, 2011.
[Crossref]

A. E. Kamal, “Network protection for mesh networks: network coding based protection using p-cycles,” IEEE/ACM Trans. Netw., vol.  18, no. 1, pp. 67–80, 2010.
[Crossref]

S. A. Aly and A. E. Kamal, “Network protection codes against link failures using network coding,” in IEEE Global Telecommunications Conf., IEEE, 2008, pp. 1–6.

S. A. Aly and A. E. Kamal, “Network coding-based protection strategy against node failures,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2009, pp. 1–5.

Kellerer, W.

A. S. Thyagaturu, A. Mercian, M. P. McGarry, M. Reisslein, and W. Kellerer, “Software defined optical networks (SDONs): a comprehensive survey,” Commun. Surv. Tutorials, vol.  18, no. 4, pp. 2738–2786, 2016.
[Crossref]

Krug, L.

Lawey, A.

A. Lawey, T. El-Gorashi, and J. Elmirghani, “BitTorrent content distribution in optical networks,” J. Lightwave Technol., vol.  32, no. 21, pp. 4209–4225, Nov.2014.
[Crossref]

A. Lawey, T. El-Gorashi, and J. Elmirghani, “Distributed energy efficient clouds over core networks,” J. Lightwave Technol., vol.  32, no. 7, pp. 1261–1281, Apr.2014.
[Crossref]

Li, S. Y. R.

R. Ahlswede, C. Ning, S. Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol.  46, no. 4, pp. 1204–1216, 2000.
[Crossref]

McGarry, M. P.

A. S. Thyagaturu, A. Mercian, M. P. McGarry, M. Reisslein, and W. Kellerer, “Software defined optical networks (SDONs): a comprehensive survey,” Commun. Surv. Tutorials, vol.  18, no. 4, pp. 2738–2786, 2016.
[Crossref]

Mercian, A.

A. S. Thyagaturu, A. Mercian, M. P. McGarry, M. Reisslein, and W. Kellerer, “Software defined optical networks (SDONs): a comprehensive survey,” Commun. Surv. Tutorials, vol.  18, no. 4, pp. 2738–2786, 2016.
[Crossref]

Mohandespour, M.

A. E. Kamal and M. Mohandespour, “Network coding-based protection,” Opt. Switching Netw., vol.  11, pp. 189–201, 2014.
[Crossref]

Muktadir, A.

A. Muktadir, A. A. Jose, and E. Oki, “An optimum mathematical programming model for network-coding based routing with 1 + 1 path protection,” in World Telecommunications Congress (WTC), IEEE, 2012, pp. 1–5.

Musa, M.

M. Musa, T. Elgorashi, and J. Elmirghani, “Energy efficient survivable IP-over-WDM networks with network coding,” J. Opt. Commun. Netw., vol.  9, no. 3, pp. 207–217, 2017.
[Crossref]

M. Musa, T. El-Gorashi, and J. Elmirghani, “Energy efficient routing and network coding assignment in core networks,” J. Lightwave Technol. (to be published).

M. Musa, T. E. El-Gorashi, and J. M. Elmirghani, “Energy efficient core networks using network coding,” in 17th Int. Conf. on Transparent Optical Networks (ICTON), IEEE, 2015, p. 14.

M. Musa, T. E. El-Gorashi, and J. M. Elmirghani, “Network coding for energy efficiency in bypass IP/WDM networks,” in 18th Int. Conf. on Transparent Optical Networks (ICTON), IEEE, 2016.

Nguyen, A. Q. T.

A. Capone, C. Cascone, A. Q. T. Nguyen, and B. Sanso, “Detour planning for fast and reliable failure recovery in SDN with OpenState,” in 11th Int. Conf. on the Design of Reliable Communication Networks (DRCN), IEEE, 2015.

Ning, C.

R. Ahlswede, C. Ning, S. Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol.  46, no. 4, pp. 1204–1216, 2000.
[Crossref]

Nonde, L.

Oki, E.

A. Muktadir, A. A. Jose, and E. Oki, “An optimum mathematical programming model for network-coding based routing with 1 + 1 path protection,” in World Telecommunications Congress (WTC), IEEE, 2012, pp. 1–5.

Osman, N.

Overby, H.

H. Overby, G. Biczok, P. Babarczi, and J. Tapolcai, “Cost comparison of 1 + 1 path protection schemes: a case for coding,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2012.

Øverby, H.

P. Babarczi, G. Biczok, H. Øverby, J. Tapolcai, and P. Soproni, “Realization strategies of dedicated path protection: a bandwidth cost perspective,” Comput. Netw., vol.  57, no. 9, pp. 1974–1990, 2013.
[Crossref]

Parthiban, R.

M. N. Dharmaweera, R. Parthiban, and Y. A. Sekercioglu, “Toward a power-efficient backbone network: the state of research,” Commun. Surv. Tutorials, vol.  17, no. 1, pp. 198–227, 2015.
[Crossref]

Polverini, M.

F. Idzikowski, L. Chiaraviglio, A. Cianfrani, J. L. Vizcano, M. Polverini, and Y. Ye, “A survey on energy-aware design and operation of core networks,” Commun. Surveys Tuts., vol.  18, no. 2, pp. 1453–1499, 2016.
[Crossref]

Rambach, F.

I. B. Barla, F. Rambach, D. A. Schupke, and M. Thakur, “Network coding for protection against multiple link failures in multi-domain networks,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2010, pp. 1–6

Reisslein, M.

A. S. Thyagaturu, A. Mercian, M. P. McGarry, M. Reisslein, and W. Kellerer, “Software defined optical networks (SDONs): a comprehensive survey,” Commun. Surv. Tutorials, vol.  18, no. 4, pp. 2738–2786, 2016.
[Crossref]

Sanso, B.

A. Capone, C. Cascone, A. Q. T. Nguyen, and B. Sanso, “Detour planning for fast and reliable failure recovery in SDN with OpenState,” in 11th Int. Conf. on the Design of Reliable Communication Networks (DRCN), IEEE, 2015.

Schupke, D. A.

I. B. Barla, F. Rambach, D. A. Schupke, and M. Thakur, “Network coding for protection against multiple link failures in multi-domain networks,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2010, pp. 1–6

Sekercioglu, Y. A.

M. N. Dharmaweera, R. Parthiban, and Y. A. Sekercioglu, “Toward a power-efficient backbone network: the state of research,” Commun. Surv. Tutorials, vol.  17, no. 1, pp. 198–227, 2015.
[Crossref]

Soproni, P.

P. Babarczi, G. Biczok, H. Øverby, J. Tapolcai, and P. Soproni, “Realization strategies of dedicated path protection: a bandwidth cost perspective,” Comput. Netw., vol.  57, no. 9, pp. 1974–1990, 2013.
[Crossref]

Tapolcai, J.

P. Babarczi, G. Biczok, H. Øverby, J. Tapolcai, and P. Soproni, “Realization strategies of dedicated path protection: a bandwidth cost perspective,” Comput. Netw., vol.  57, no. 9, pp. 1974–1990, 2013.
[Crossref]

H. Overby, G. Biczok, P. Babarczi, and J. Tapolcai, “Cost comparison of 1 + 1 path protection schemes: a case for coding,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2012.

Thakur, M.

I. B. Barla, F. Rambach, D. A. Schupke, and M. Thakur, “Network coding for protection against multiple link failures in multi-domain networks,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2010, pp. 1–6

Thyagaturu, A. S.

A. S. Thyagaturu, A. Mercian, M. P. McGarry, M. Reisslein, and W. Kellerer, “Software defined optical networks (SDONs): a comprehensive survey,” Commun. Surv. Tutorials, vol.  18, no. 4, pp. 2738–2786, 2016.
[Crossref]

Vizcano, J. L.

F. Idzikowski, L. Chiaraviglio, A. Cianfrani, J. L. Vizcano, M. Polverini, and Y. Ye, “A survey on energy-aware design and operation of core networks,” Commun. Surveys Tuts., vol.  18, no. 2, pp. 1453–1499, 2016.
[Crossref]

Ye, Y.

F. Idzikowski, L. Chiaraviglio, A. Cianfrani, J. L. Vizcano, M. Polverini, and Y. Ye, “A survey on energy-aware design and operation of core networks,” Commun. Surveys Tuts., vol.  18, no. 2, pp. 1453–1499, 2016.
[Crossref]

Yeung, R. W.

R. Ahlswede, C. Ning, S. Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol.  46, no. 4, pp. 1204–1216, 2000.
[Crossref]

Commun. Surv. Tutorials (2)

M. N. Dharmaweera, R. Parthiban, and Y. A. Sekercioglu, “Toward a power-efficient backbone network: the state of research,” Commun. Surv. Tutorials, vol.  17, no. 1, pp. 198–227, 2015.
[Crossref]

A. S. Thyagaturu, A. Mercian, M. P. McGarry, M. Reisslein, and W. Kellerer, “Software defined optical networks (SDONs): a comprehensive survey,” Commun. Surv. Tutorials, vol.  18, no. 4, pp. 2738–2786, 2016.
[Crossref]

Commun. Surveys Tuts. (1)

F. Idzikowski, L. Chiaraviglio, A. Cianfrani, J. L. Vizcano, M. Polverini, and Y. Ye, “A survey on energy-aware design and operation of core networks,” Commun. Surveys Tuts., vol.  18, no. 2, pp. 1453–1499, 2016.
[Crossref]

Comput. Netw. (1)

P. Babarczi, G. Biczok, H. Øverby, J. Tapolcai, and P. Soproni, “Realization strategies of dedicated path protection: a bandwidth cost perspective,” Comput. Netw., vol.  57, no. 9, pp. 1974–1990, 2013.
[Crossref]

IEEE Trans. Commun. (1)

A. E. Kamal and O. Al-Kofahi, “Efficient and agile 1 + n protection,” IEEE Trans. Commun., vol.  59, no. 1, pp. 169–180, 2011.
[Crossref]

IEEE Trans. Inf. Theory (1)

R. Ahlswede, C. Ning, S. Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol.  46, no. 4, pp. 1204–1216, 2000.
[Crossref]

IEEE/ACM Trans. Netw. (1)

A. E. Kamal, “Network protection for mesh networks: network coding based protection using p-cycles,” IEEE/ACM Trans. Netw., vol.  18, no. 1, pp. 67–80, 2010.
[Crossref]

J. Lightwave Technol. (7)

J. Opt. Commun. Netw. (1)

Opt. Switching Netw. (1)

A. E. Kamal and M. Mohandespour, “Network coding-based protection,” Opt. Switching Netw., vol.  11, pp. 189–201, 2014.
[Crossref]

Other (10)

M. Musa, T. El-Gorashi, and J. Elmirghani, “Energy efficient routing and network coding assignment in core networks,” J. Lightwave Technol. (to be published).

S. A. Aly and A. E. Kamal, “Network protection codes against link failures using network coding,” in IEEE Global Telecommunications Conf., IEEE, 2008, pp. 1–6.

S. A. Aly and A. E. Kamal, “Network coding-based protection strategy against node failures,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2009, pp. 1–5.

I. B. Barla, F. Rambach, D. A. Schupke, and M. Thakur, “Network coding for protection against multiple link failures in multi-domain networks,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2010, pp. 1–6

A. Muktadir, A. A. Jose, and E. Oki, “An optimum mathematical programming model for network-coding based routing with 1 + 1 path protection,” in World Telecommunications Congress (WTC), IEEE, 2012, pp. 1–5.

GreenTouch, “Reducing the Net Energy Consumption in Communications Networks by up to 98% by 2020,” [Online]. Available: https://s3-us-west-2.amazonaws.com/belllabs-microsite-greentouch/uploads/documents/GreenTouch_Green_Meter_Final_Results_18_June_2015.pdf .

H. Overby, G. Biczok, P. Babarczi, and J. Tapolcai, “Cost comparison of 1 + 1 path protection schemes: a case for coding,” in IEEE Int. Conf. on Communications (ICC), IEEE, 2012.

M. Musa, T. E. El-Gorashi, and J. M. Elmirghani, “Energy efficient core networks using network coding,” in 17th Int. Conf. on Transparent Optical Networks (ICTON), IEEE, 2015, p. 14.

M. Musa, T. E. El-Gorashi, and J. M. Elmirghani, “Network coding for energy efficiency in bypass IP/WDM networks,” in 18th Int. Conf. on Transparent Optical Networks (ICTON), IEEE, 2016.

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Figures (8)

Fig. 1.
Fig. 1. Example of using network coding for protection.
Fig. 2.
Fig. 2. Weighted encodable graph for a full mesh of size 4, 5, and 6, respectively.
Fig. 3.
Fig. 3. Power consumption of the five-node full mesh topology with equal demands using the MILP, heuristic, and analytical bound.
Fig. 4.
Fig. 4. Power savings of the full mesh topology under different network sizes.
Fig. 5.
Fig. 5. Encoding pairs for two rings with different odd numbers of nodes.
Fig. 6.
Fig. 6. Encoding pairs for two rings with different even numbers of nodes.
Fig. 7.
Fig. 7. Power consumption of the five-node ring topology with equal demands using the MILP, heuristic, and analytical bound.
Fig. 8.
Fig. 8. Power savings of the ring topology under different network sizes.

Equations (70)

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P=(pp+ptB)dDm,nVd(xmnd+ymnd)(pp+ptB)d1,d2m,nmin(Vd1,Vd2)βmnd1d22,
P1=(pp+ptB)(dDm,nVd(xmnd+ymnd)),
P2=(pp+ptB)(d1,d2m,nmin(Vd1,Vd2)βmnd1d22).
P1=(pp+ptB)(dDVdm,n(xmnd+ymnd)).
m,n(xmnd+ymnd)2hmind.
P1(pp+ptB)(dD2Vdhmind).
P1(pp+ptB)hmin(dD2Vd),
P12(pp+ptB)N(N1)Vhmin.
P02(pp+ptB)N(N1)Vhmin.
min(Vd1,Vd2)Vd1+Vd22.
P2(pp+ptB)(d1,d2m,nVd1+Vd22βmnd1d22).
P2pp+pt2B(d1,d2m,nVd1,d2βmnd1d2).
P2pp+pt2B(d1,d2Vd1,d2m,nβmnd1d2).
hd2d1=m,nβmnd1d2.
P2pp+pt2B(d1,d2Vd1,d2hd2d1).
P(pp+ptB)(2dDVdhmind12d1,d2Vd1,d2hd2d1).
P(pp+ptB)dDVd(2hmind12d2hd2d).
d2Dbd2d11.
P(pp+ptB)(dDVd(2hmindh^d2)),
P2(pp+ptB)(dDVd(hmindh^d4)).
h˜d=hmindh^d4,
P2(pp+ptB)(dDVdh˜d).
1nk=1nakbk(1nk=1nak)(1nk=1nbk),
P2(pp+ptB)(1N(N1)dDVddDh˜d),
P2(pp+ptB)VdDh˜d,
P2(pp+ptB)N(N1)Vh˜.
P0=(pp+ptB)(dDm,nVd(xmnd+ymnd)).
P0=(pp+ptB)(dD3Vd),
P0=3(pp+ptB)VN(N1).
P=P0(pp+ptB)d1,d2m,nmin(Vd1,Vd2)βmnd1d22,
P=Po(pp+ptB)V2d1,d2m,nβmnd1d2,
P=Po(pp+ptB)V2d1,d2m,nβmnd1d2.
d1,d2m,nβmnd1d2=N(N1).
P=(pp+ptB)N(N1)5V2.
ϕodd=1(pp+ptB)N(N1)5V23(pp+ptB)VN(N1)=0.166,
d1,d2m,nβmnd1d2=N(N2).
P=3(pp+ptB)VN(N1)(pp+ptB)VN(N2)2,
P=(pp+ptB)VN(5N42).
ϕeven=1(pp+ptB)VN(5N42)3(pp+ptB)VN(N1),
ϕeven=N26(N1).
ε(N)=16N26(N1)=16(N1),
limNε(N)=limN16(N1)=0.
P0=(pp+ptB)(dDm,nVd(xmnd+ymnd)).
hw=dDm,nxmnd=2N(1+2++(N12))=(N1)N(N+1)4.
hp=dDm,nymnd=2N(N1+N2++NN12)=N(N1)(3N14),
dDm,n(xmnd+ymnd)=(N1)N(N+1)4+N(N1)(3N14)=N3N2.
hw=dDm,nxmnd=2N(1+2++(N22))+NN2=N34.
hp=dDm,nymnd=2N(N1+N2++NN22)+NN2,
hp=2N(N(N22)12N22)+N22=N2(3N4)4,
dDm,n(xmnd+ymnd)=N34+N2(3N4)4=N3N2.
ht(odd1)=2N[(N2)+(N4)++(NN32)].
ht(odd1)=2N[N(N34)24N32],
ht(odd1)=2N[N(N34)2k=1N34k]=N(N3)(3N1)8.
ϕodd1=N(N3)(3N1)8(N3N2),
limNϕodd1=limNN(N3)(3N1)8(N3N2)=38=37.5%.
ht(odd2)=2N[(N2)+(N4)++(NN12)],
ht(odd2)=2N[NN1424N12],
ht(odd2)=2N[NN142k=1N14k]=38N(N1)2.
ϕodd2=38N(N1)2N3N2,
limnϕodd2=limn38N(N1)2N3N2=38=37.5%.
ht(even1)=N(2[NN4424N42]+N2).
ht(even1)=N(2[NN442k=1N44k]+N2),
ht(even1)=N22[1+3(N4)4].
ϕeven1=12N2(3N42)N3N2;
limNϕeven1=limN12N2(3N42)N3N2=38=37.5%.
ht(even2)=N(2[NN2424N22]),
ht(even2)=N(2[NN242k=1N24k]),
ht(even2)=N(N2)(3N2)8.
ϕeven2=18N(N2)(3N2)N3N2,
limNϕeven2=limN18N(N2)(3N2)N3N2=38=37.5%.