Abstract

The deployment of translucent optical networks is considered the most promising short term solution to decrease costs and energy consumption in optical backbone networks. In fact, translucent wavelength switched optical networks (WSONs) have recently received great attention from the research community due to their technological maturity. However, the inflexibility and coarse granularity of WSONs is (re-)fostering research interest in sub-wavelength switching technologies such as optical burst switching (OBS). In OBS, however, the majority of research works neglect the impact of physical layer impairments by considering either fully transparent (i.e., with optical 3R regeneration) or opaque (i.e., with electrical 3R regeneration) networks. For this very reason, in this paper we present a translucent OBS (T-OBS) network architecture which aims at bridging the gap between the transparent and opaque solutions. In the T-OBS network the problem of routing and regenerator placement and dimensioning (RRPD) emerges. Joint RRPD is a complex problem and, in order to approach it, we propose to decompose it into the routing and RPD subproblems. As a consequence, we provide a mixed integer linear programming formulation of the routing problem and several heuristic strategies for the RPD problem. Illustrative numerical results prove the effectiveness of these methods at minimizing the number of electrical 3R regenerators deployed in the network. Considering a broad range of network topologies, we show that the proposed RPD heuristics ensure a proper quality of transmission performance whilst at the same time providing a cost-effective T-OBS network architecture.

© 2011 OSA

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2011

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global. Optim. 50, (3), 503‒518 (2011).
[CrossRef]

R. Reis, M. Ritt, L. Buriol, and M. Resende, "A biased random-key genetic algorithm for OSPF and DEFT routing to minimize network congestion," Int. Trans. Oper. Res. 18, (3), 401‒423 (2011).

2010

S. Orlowski, M. Pióro, A. Tomaszewski, and R. Wessäly, "SNDlib 1.0 survivable network design library," Networks 55, (3), 276‒286 (2010).

J. Triay and C. Cervelló-Pastor, "An ant-based algorithm for distributed routing and wavelength assignment in dynamic optical networks," J. Sel. Areas Commun. 28, (4), 542‒552 (2010).
[CrossRef]

O. Pedrola, S. Rumley, M. Klinkowski, D. Careglio, C. Gaumier, and J. Solé-Pareta, "JAVOBS: a flexible simulator for OBS network architectures," J. Netw. 5, (2), 256‒264 (2010).

K. Manousakis, P. Kokkinos, K. Christodoulopoulos, and E. Varvarigos, "Joint online routing, wavelength assignment and regenerator allocation in translucent optical networks," J. Lightwave Technol. 28, (8), 1152‒1163 (2010).
[CrossRef]

B. G. Bathula, R. R. C. Bikram, V. M. Vokkarane, and S. Talabattula, "Quality of transmission aware manycasting over optical burst-switched (OBS) networks," J. Opt. Commun. Netw. 2, (10), 820‒829 (2010).
[CrossRef]

R. Martínez, R. Casellas, R. Muñoz, and T. Tsuritani, "Experimental translucent-oriented routing for dynamic lightpath provisioning in GMPLS-enabled wavelength switched optical networks," J. Lightwave Technol. 28, (8), 1241‒1255 (2010).
[CrossRef]

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

2009

S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Solé-Pareta, and I. Tomkos, "A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks," Comput. Netw. 53, (7), 926‒944 (2009).
[CrossRef]

H. Pereira, D. A. R. Chaves, C. J. A. Bastos-Filho, and J. F. Martins-Filho, "OSNR model to consider physical layer impairments in transparent optical networks," Photonic Network Commun. 18, (2), 137‒149 (2009).
[CrossRef]

Y. Fan and B. Wang, "Physical impairment aware scheduling in optical burst switched networks," Photonic Network Commun. 18, (2), 244‒254 (2009).
[CrossRef]

R. Muñoz, R. Martínez, and R. Casellas, "Challenges for GMPLS lightpath provisioning in transparent optical networks: wavelength constraints in routing and signalling," IEEE Commun. Mag. 47, (8), 26‒34 (2009).

M. Klinkowski, D. Careglio, J. Solé-Pareta, and M. Marciniak, "Performance overview of the offset time emulated OBS network architecture," J. Lightwave Technol. 27, (14), 2751‒2764 (2009).
[CrossRef]

2008

2007

G. Shen and R. S. Tucker, "Translucent optical networks: the way forward," IEEE Commun. Mag. 45, (2), 48‒54 (2007).
[CrossRef]

2006

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

S. Ngo, X. Jiang, and S. Horiguchi, "An ant-based approach for dynamic RWA in optical WDM networks," Photonic Network Commun. 11, (1), 39‒48 (2006).
[CrossRef]

2005

M. Dorigo and C. Blum, "Ant colony optimization theory: A survey," Theor. Comput. Sci. 344, (2–3), 243‒278 (2005).
[CrossRef]

X. Yang and B. Ramamurthy, "Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing," Photonic Network Commun. 10, (1), 39‒53 (2005).
[CrossRef]

2003

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, "Performance analyses of optical burst-switching networks," IEEE J. Sel. Areas Commun. 21, (7), 1187‒1197 (2003).
[CrossRef]

2002

R. Garlick and R. Barr, "Dynamic wavelength routing in WDM networks via ant colony optimization," Lect. Notes Comput. Sci. 2463, 27‒41 (2002).

2000

1999

C. Qiao and M. Yoo, "Optical burst switching (OBS)–A new paradigm for an optical Internet," J. High Speed Netw. 8, (1), 69‒84 (1999).

Andriolli, N.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Azodolmolky, S.

S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Solé-Pareta, and I. Tomkos, "A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks," Comput. Netw. 53, (7), 926‒944 (2009).
[CrossRef]

Bandyopadhyay, S.

A. Sen, S. Murthy, and S. Bandyopadhyay, "On sparse placement of regenerator nodes in translucent optical networks," Proc. IEEE Globecom 2008, Nov. 2008, pp. 1‒6.

Barr, R.

R. Garlick and R. Barr, "Dynamic wavelength routing in WDM networks via ant colony optimization," Lect. Notes Comput. Sci. 2463, 27‒41 (2002).

Bastos-Filho, C. J. A.

H. Pereira, D. A. R. Chaves, C. J. A. Bastos-Filho, and J. F. Martins-Filho, "OSNR model to consider physical layer impairments in transparent optical networks," Photonic Network Commun. 18, (2), 137‒149 (2009).
[CrossRef]

Bathula, B. G.

Bernstein, G.

Y. Lee, G. Bernstein, D. Li, and G. Martinelli, "A framework for the control of wavelength switched optical networks (WSON) with impairments," IETF Internet draft, Oct. 2010.

Bikram, R. R. C.

Blum, C.

M. Dorigo and C. Blum, "Ant colony optimization theory: A survey," Theor. Comput. Sci. 344, (2–3), 243‒278 (2005).
[CrossRef]

Buchta, H.

Buriol, L.

R. Reis, M. Ritt, L. Buriol, and M. Resende, "A biased random-key genetic algorithm for OSPF and DEFT routing to minimize network congestion," Int. Trans. Oper. Res. 18, (3), 401‒423 (2011).

Careglio, D.

O. Pedrola, S. Rumley, M. Klinkowski, D. Careglio, C. Gaumier, and J. Solé-Pareta, "JAVOBS: a flexible simulator for OBS network architectures," J. Netw. 5, (2), 256‒264 (2010).

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

M. Klinkowski, D. Careglio, J. Solé-Pareta, and M. Marciniak, "Performance overview of the offset time emulated OBS network architecture," J. Lightwave Technol. 27, (14), 2751‒2764 (2009).
[CrossRef]

S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Solé-Pareta, and I. Tomkos, "A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks," Comput. Netw. 53, (7), 926‒944 (2009).
[CrossRef]

O. Pedrola, D. Careglio, M. Klinkowski, and J. Solé-Pareta, "RRPD strategies for a T-OBS network architecture," Proc. IEEE HPSR 2011, July 2011, Cartagena, Spain, pp. 95‒100.

O. Pedrola, D. Careglio, M. Klinkowski, and J. Solé-Pareta, "Modelling and performance evaluation of a translucent OBS network architecture," Proc. IEEE Globecom 2010, Dec. 2010, Miami, FL, pp. 1‒6.

Casellas, R.

R. Martínez, R. Casellas, R. Muñoz, and T. Tsuritani, "Experimental translucent-oriented routing for dynamic lightpath provisioning in GMPLS-enabled wavelength switched optical networks," J. Lightwave Technol. 28, (8), 1241‒1255 (2010).
[CrossRef]

R. Muñoz, R. Martínez, and R. Casellas, "Challenges for GMPLS lightpath provisioning in transparent optical networks: wavelength constraints in routing and signalling," IEEE Commun. Mag. 47, (8), 26‒34 (2009).

Castoldi, P.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Cervelló-Pastor, C.

J. Triay and C. Cervelló-Pastor, "An ant-based algorithm for distributed routing and wavelength assignment in dynamic optical networks," J. Sel. Areas Commun. 28, (4), 542‒552 (2010).
[CrossRef]

Chaves, D. A. R.

H. Pereira, D. A. R. Chaves, C. J. A. Bastos-Filho, and J. F. Martins-Filho, "OSNR model to consider physical layer impairments in transparent optical networks," Photonic Network Commun. 18, (2), 137‒149 (2009).
[CrossRef]

Christodoulopoulos, K.

Colle, D.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Comellas, J.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Cugini, F.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Datta, D.

B. Ramamurthy, H. Fenq, D. Datta, J. P. Heritage, and B. Mukherjee, "Transparent vs. opaque vs. translucent wavelength-routed optical networks," Proc. IEEE/OSA OFC’1999, Vol. 1, Feb. 1999, San Diego, CA, pp. 59‒61.

De Maesschalck, S.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Demeester, P.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Derkacz, J.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Djupsjobacka, A.

M. Mestre, J. M. Fabrega, J. A. Lazaro, V. Polo, A. Djupsjobacka, M. Forzati, P. J. Rigole, and J. Prat, "Tuning characteristics and switching speed of a modulated grating Y structure laser for wavelength routed PONs," Proc. ANIC 2010, June 2010, Karlsrhue, Germany, pp. 1‒2.

Dorigo, M.

M. Dorigo and C. Blum, "Ant colony optimization theory: A survey," Theor. Comput. Sci. 344, (2–3), 243‒278 (2005).
[CrossRef]

M. Dorigo and T. Stützle, Ant Colony Optimization, The MIT Press, Cambridge, MA, 2004.

Fabrega, J. M.

M. Mestre, J. M. Fabrega, J. A. Lazaro, V. Polo, A. Djupsjobacka, M. Forzati, P. J. Rigole, and J. Prat, "Tuning characteristics and switching speed of a modulated grating Y structure laser for wavelength routed PONs," Proc. ANIC 2010, June 2010, Karlsrhue, Germany, pp. 1‒2.

Fan, Y.

Y. Fan and B. Wang, "Physical impairment aware scheduling in optical burst switched networks," Photonic Network Commun. 18, (2), 244‒254 (2009).
[CrossRef]

Fenq, H.

B. Ramamurthy, H. Fenq, D. Datta, J. P. Heritage, and B. Mukherjee, "Transparent vs. opaque vs. translucent wavelength-routed optical networks," Proc. IEEE/OSA OFC’1999, Vol. 1, Feb. 1999, San Diego, CA, pp. 59‒61.

Forzati, M.

M. Mestre, J. M. Fabrega, J. A. Lazaro, V. Polo, A. Djupsjobacka, M. Forzati, P. J. Rigole, and J. Prat, "Tuning characteristics and switching speed of a modulated grating Y structure laser for wavelength routed PONs," Proc. ANIC 2010, June 2010, Karlsrhue, Germany, pp. 1‒2.

Garlick, R.

R. Garlick and R. Barr, "Dynamic wavelength routing in WDM networks via ant colony optimization," Lect. Notes Comput. Sci. 2463, 27‒41 (2002).

Gauger, C. M.

Gaumier, C.

O. Pedrola, S. Rumley, M. Klinkowski, D. Careglio, C. Gaumier, and J. Solé-Pareta, "JAVOBS: a flexible simulator for OBS network architectures," J. Netw. 5, (2), 256‒264 (2010).

Gonçalves, J.

J. Gonçalves and M. Resende, "Biased random-key genetic algorithms for combinatorial optimization," J. Heuristics (posted 27 August 2010, in press).

Heritage, J. P.

B. Ramamurthy, H. Fenq, D. Datta, J. P. Heritage, and B. Mukherjee, "Transparent vs. opaque vs. translucent wavelength-routed optical networks," Proc. IEEE/OSA OFC’1999, Vol. 1, Feb. 1999, San Diego, CA, pp. 59‒61.

Horiguchi, S.

S. Ngo, X. Jiang, and S. Horiguchi, "An ant-based approach for dynamic RWA in optical WDM networks," Photonic Network Commun. 11, (1), 39‒48 (2006).
[CrossRef]

Inkret, R.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Jaeger, M.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Jiang, X.

S. Ngo, X. Jiang, and S. Horiguchi, "An ant-based approach for dynamic RWA in optical WDM networks," Photonic Network Commun. 11, (1), 39‒48 (2006).
[CrossRef]

Junyent, G.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Kashihara, S.

T. Tsuritani, M. Miyazawa, S. Kashihara, and T. Otani, "Optical path computation element interworking with network management system for transparent mesh networks," Proc. IEEE/OSA OFC/NFOEC 2008, Feb. 2008, San Diego, CA, pp. 1‒10.

Kleinberg, J.

J. Kleinberg and E. Tardos, Algorithm Design, Addison-Wesley, 2005, pp. 661‒706.

Klinkowski, M.

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

O. Pedrola, S. Rumley, M. Klinkowski, D. Careglio, C. Gaumier, and J. Solé-Pareta, "JAVOBS: a flexible simulator for OBS network architectures," J. Netw. 5, (2), 256‒264 (2010).

M. Klinkowski, D. Careglio, J. Solé-Pareta, and M. Marciniak, "Performance overview of the offset time emulated OBS network architecture," J. Lightwave Technol. 27, (14), 2751‒2764 (2009).
[CrossRef]

S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Solé-Pareta, and I. Tomkos, "A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks," Comput. Netw. 53, (7), 926‒944 (2009).
[CrossRef]

O. Pedrola, D. Careglio, M. Klinkowski, and J. Solé-Pareta, "RRPD strategies for a T-OBS network architecture," Proc. IEEE HPSR 2011, July 2011, Cartagena, Spain, pp. 95‒100.

O. Pedrola, D. Careglio, M. Klinkowski, and J. Solé-Pareta, "Modelling and performance evaluation of a translucent OBS network architecture," Proc. IEEE Globecom 2010, Dec. 2010, Miami, FL, pp. 1‒6.

Kokkinos, P.

Krummrich, P. M.

S. Pachnique, T. Paschenda, and P. M. Krummrich, "Physical impairment based regenerator placement and routing in translucent optical networks," Proc. IEEE/OSA OFC/NFOEC 2008, Feb. 2008, San Diego, CA, pp. 1‒3.

Lazaro, J. A.

M. Mestre, J. M. Fabrega, J. A. Lazaro, V. Polo, A. Djupsjobacka, M. Forzati, P. J. Rigole, and J. Prat, "Tuning characteristics and switching speed of a modulated grating Y structure laser for wavelength routed PONs," Proc. ANIC 2010, June 2010, Karlsrhue, Germany, pp. 1‒2.

Lee, Y.

Y. Lee, G. Bernstein, D. Li, and G. Martinelli, "A framework for the control of wavelength switched optical networks (WSON) with impairments," IETF Internet draft, Oct. 2010.

Li, D.

Y. Lee, G. Bernstein, D. Li, and G. Martinelli, "A framework for the control of wavelength switched optical networks (WSON) with impairments," IETF Internet draft, Oct. 2010.

Lievens, I.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Manousakis, K.

Marciniak, M.

Marin, E.

S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Solé-Pareta, and I. Tomkos, "A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks," Comput. Netw. 53, (7), 926‒944 (2009).
[CrossRef]

Martinelli, G.

Y. Lee, G. Bernstein, D. Li, and G. Martinelli, "A framework for the control of wavelength switched optical networks (WSON) with impairments," IETF Internet draft, Oct. 2010.

Martínez, R.

R. Martínez, R. Casellas, R. Muñoz, and T. Tsuritani, "Experimental translucent-oriented routing for dynamic lightpath provisioning in GMPLS-enabled wavelength switched optical networks," J. Lightwave Technol. 28, (8), 1241‒1255 (2010).
[CrossRef]

R. Muñoz, R. Martínez, and R. Casellas, "Challenges for GMPLS lightpath provisioning in transparent optical networks: wavelength constraints in routing and signalling," IEEE Commun. Mag. 47, (8), 26‒34 (2009).

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Martins-Filho, J. F.

H. Pereira, D. A. R. Chaves, C. J. A. Bastos-Filho, and J. F. Martins-Filho, "OSNR model to consider physical layer impairments in transparent optical networks," Photonic Network Commun. 18, (2), 137‒149 (2009).
[CrossRef]

Mauz, C.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

McFarland, R. I.

Mestre, M.

M. Mestre, J. M. Fabrega, J. A. Lazaro, V. Polo, A. Djupsjobacka, M. Forzati, P. J. Rigole, and J. Prat, "Tuning characteristics and switching speed of a modulated grating Y structure laser for wavelength routed PONs," Proc. ANIC 2010, June 2010, Karlsrhue, Germany, pp. 1‒2.

Mikac, B.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Miyazawa, M.

T. Tsuritani, M. Miyazawa, S. Kashihara, and T. Otani, "Optical path computation element interworking with network management system for transparent mesh networks," Proc. IEEE/OSA OFC/NFOEC 2008, Feb. 2008, San Diego, CA, pp. 1‒10.

Monteiro, P.

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

Mukherjee, B.

B. Ramamurthy, H. Fenq, D. Datta, J. P. Heritage, and B. Mukherjee, "Transparent vs. opaque vs. translucent wavelength-routed optical networks," Proc. IEEE/OSA OFC’1999, Vol. 1, Feb. 1999, San Diego, CA, pp. 59‒61.

Muñoz, R.

R. Martínez, R. Casellas, R. Muñoz, and T. Tsuritani, "Experimental translucent-oriented routing for dynamic lightpath provisioning in GMPLS-enabled wavelength switched optical networks," J. Lightwave Technol. 28, (8), 1241‒1255 (2010).
[CrossRef]

R. Muñoz, R. Martínez, and R. Casellas, "Challenges for GMPLS lightpath provisioning in transparent optical networks: wavelength constraints in routing and signalling," IEEE Commun. Mag. 47, (8), 26‒34 (2009).

Murthy, S.

A. Sen, S. Murthy, and S. Bandyopadhyay, "On sparse placement of regenerator nodes in translucent optical networks," Proc. IEEE Globecom 2008, Nov. 2008, pp. 1‒6.

Ngo, S.

S. Ngo, X. Jiang, and S. Horiguchi, "An ant-based approach for dynamic RWA in optical WDM networks," Photonic Network Commun. 11, (1), 39‒48 (2006).
[CrossRef]

Noronha, T.

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global. Optim. 50, (3), 503‒518 (2011).
[CrossRef]

Nygard, K. E.

W. Zhang, J. Tang, K. E. Nygard, and C. Wang, "REPARE: Regenerator placement and routing establishment in translucent networks," Proc. IEEE Globecom 2009, Nov. 2009, pp. 1‒7.

Orlowski, S.

S. Orlowski, M. Pióro, A. Tomaszewski, and R. Wessäly, "SNDlib 1.0 survivable network design library," Networks 55, (3), 276‒286 (2010).

Otani, T.

T. Tsuritani, M. Miyazawa, S. Kashihara, and T. Otani, "Optical path computation element interworking with network management system for transparent mesh networks," Proc. IEEE/OSA OFC/NFOEC 2008, Feb. 2008, San Diego, CA, pp. 1‒10.

Pachnique, S.

S. Pachnique, T. Paschenda, and P. M. Krummrich, "Physical impairment based regenerator placement and routing in translucent optical networks," Proc. IEEE/OSA OFC/NFOEC 2008, Feb. 2008, San Diego, CA, pp. 1‒3.

Paschenda, T.

S. Pachnique, T. Paschenda, and P. M. Krummrich, "Physical impairment based regenerator placement and routing in translucent optical networks," Proc. IEEE/OSA OFC/NFOEC 2008, Feb. 2008, San Diego, CA, pp. 1‒3.

Patzak, E.

Pedro, J.

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

Pedrola, O.

O. Pedrola, S. Rumley, M. Klinkowski, D. Careglio, C. Gaumier, and J. Solé-Pareta, "JAVOBS: a flexible simulator for OBS network architectures," J. Netw. 5, (2), 256‒264 (2010).

O. Pedrola, D. Careglio, M. Klinkowski, and J. Solé-Pareta, "RRPD strategies for a T-OBS network architecture," Proc. IEEE HPSR 2011, July 2011, Cartagena, Spain, pp. 95‒100.

O. Pedrola, D. Careglio, M. Klinkowski, and J. Solé-Pareta, "Modelling and performance evaluation of a translucent OBS network architecture," Proc. IEEE Globecom 2010, Dec. 2010, Miami, FL, pp. 1‒6.

Pereira, H.

H. Pereira, D. A. R. Chaves, C. J. A. Bastos-Filho, and J. F. Martins-Filho, "OSNR model to consider physical layer impairments in transparent optical networks," Photonic Network Commun. 18, (2), 137‒149 (2009).
[CrossRef]

Pickavet, M.

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Pinart, C.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Pióro, M.

S. Orlowski, M. Pióro, A. Tomaszewski, and R. Wessäly, "SNDlib 1.0 survivable network design library," Networks 55, (3), 276‒286 (2010).

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

Pires, J.

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

Polo, V.

M. Mestre, J. M. Fabrega, J. A. Lazaro, V. Polo, A. Djupsjobacka, M. Forzati, P. J. Rigole, and J. Prat, "Tuning characteristics and switching speed of a modulated grating Y structure laser for wavelength routed PONs," Proc. ANIC 2010, June 2010, Karlsrhue, Germany, pp. 1‒2.

Prat, J.

M. Mestre, J. M. Fabrega, J. A. Lazaro, V. Polo, A. Djupsjobacka, M. Forzati, P. J. Rigole, and J. Prat, "Tuning characteristics and switching speed of a modulated grating Y structure laser for wavelength routed PONs," Proc. ANIC 2010, June 2010, Karlsrhue, Germany, pp. 1‒2.

Qiao, C.

C. Qiao and M. Yoo, "Optical burst switching (OBS)–A new paradigm for an optical Internet," J. High Speed Netw. 8, (1), 69‒84 (1999).

Ramamurthy, B.

X. Yang and B. Ramamurthy, "Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing," Photonic Network Commun. 10, (1), 39‒53 (2005).
[CrossRef]

B. Ramamurthy, H. Fenq, D. Datta, J. P. Heritage, and B. Mukherjee, "Transparent vs. opaque vs. translucent wavelength-routed optical networks," Proc. IEEE/OSA OFC’1999, Vol. 1, Feb. 1999, San Diego, CA, pp. 59‒61.

Reis, R.

R. Reis, M. Ritt, L. Buriol, and M. Resende, "A biased random-key genetic algorithm for OSPF and DEFT routing to minimize network congestion," Int. Trans. Oper. Res. 18, (3), 401‒423 (2011).

Resende, M.

R. Reis, M. Ritt, L. Buriol, and M. Resende, "A biased random-key genetic algorithm for OSPF and DEFT routing to minimize network congestion," Int. Trans. Oper. Res. 18, (3), 401‒423 (2011).

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global. Optim. 50, (3), 503‒518 (2011).
[CrossRef]

J. Gonçalves and M. Resende, "Biased random-key genetic algorithms for combinatorial optimization," J. Heuristics (posted 27 August 2010, in press).

Ribeiro, C.

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global. Optim. 50, (3), 503‒518 (2011).
[CrossRef]

Rigole, P. J.

M. Mestre, J. M. Fabrega, J. A. Lazaro, V. Polo, A. Djupsjobacka, M. Forzati, P. J. Rigole, and J. Prat, "Tuning characteristics and switching speed of a modulated grating Y structure laser for wavelength routed PONs," Proc. ANIC 2010, June 2010, Karlsrhue, Germany, pp. 1‒2.

Ritt, M.

R. Reis, M. Ritt, L. Buriol, and M. Resende, "A biased random-key genetic algorithm for OSPF and DEFT routing to minimize network congestion," Int. Trans. Oper. Res. 18, (3), 401‒423 (2011).

Rosberg, Z.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, "Performance analyses of optical burst-switching networks," IEEE J. Sel. Areas Commun. 21, (7), 1187‒1197 (2003).
[CrossRef]

Rumley, S.

O. Pedrola, S. Rumley, M. Klinkowski, D. Careglio, C. Gaumier, and J. Solé-Pareta, "JAVOBS: a flexible simulator for OBS network architectures," J. Netw. 5, (2), 256‒264 (2010).

Sen, A.

A. Sen, S. Murthy, and S. Bandyopadhyay, "On sparse placement of regenerator nodes in translucent optical networks," Proc. IEEE Globecom 2008, Nov. 2008, pp. 1‒6.

Shen, G.

G. Shen and R. S. Tucker, "Translucent optical networks: the way forward," IEEE Commun. Mag. 45, (2), 48‒54 (2007).
[CrossRef]

Solé-Pareta, J.

O. Pedrola, S. Rumley, M. Klinkowski, D. Careglio, C. Gaumier, and J. Solé-Pareta, "JAVOBS: a flexible simulator for OBS network architectures," J. Netw. 5, (2), 256‒264 (2010).

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

M. Klinkowski, D. Careglio, J. Solé-Pareta, and M. Marciniak, "Performance overview of the offset time emulated OBS network architecture," J. Lightwave Technol. 27, (14), 2751‒2764 (2009).
[CrossRef]

S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Solé-Pareta, and I. Tomkos, "A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks," Comput. Netw. 53, (7), 926‒944 (2009).
[CrossRef]

O. Pedrola, D. Careglio, M. Klinkowski, and J. Solé-Pareta, "RRPD strategies for a T-OBS network architecture," Proc. IEEE HPSR 2011, July 2011, Cartagena, Spain, pp. 95‒100.

O. Pedrola, D. Careglio, M. Klinkowski, and J. Solé-Pareta, "Modelling and performance evaluation of a translucent OBS network architecture," Proc. IEEE Globecom 2010, Dec. 2010, Miami, FL, pp. 1‒6.

Stützle, T.

M. Dorigo and T. Stützle, Ant Colony Optimization, The MIT Press, Cambridge, MA, 2004.

Talabattula, S.

Tang, J.

W. Zhang, J. Tang, K. E. Nygard, and C. Wang, "REPARE: Regenerator placement and routing establishment in translucent networks," Proc. IEEE Globecom 2009, Nov. 2009, pp. 1‒7.

Tardos, E.

J. Kleinberg and E. Tardos, Algorithm Design, Addison-Wesley, 2005, pp. 661‒706.

Tomaszewski, A.

S. Orlowski, M. Pióro, A. Tomaszewski, and R. Wessäly, "SNDlib 1.0 survivable network design library," Networks 55, (3), 276‒286 (2010).

Tomkos, I.

S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Solé-Pareta, and I. Tomkos, "A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks," Comput. Netw. 53, (7), 926‒944 (2009).
[CrossRef]

Triay, J.

J. Triay and C. Cervelló-Pastor, "An ant-based algorithm for distributed routing and wavelength assignment in dynamic optical networks," J. Sel. Areas Commun. 28, (4), 542‒552 (2010).
[CrossRef]

Tsuritani, T.

R. Martínez, R. Casellas, R. Muñoz, and T. Tsuritani, "Experimental translucent-oriented routing for dynamic lightpath provisioning in GMPLS-enabled wavelength switched optical networks," J. Lightwave Technol. 28, (8), 1241‒1255 (2010).
[CrossRef]

T. Tsuritani, M. Miyazawa, S. Kashihara, and T. Otani, "Optical path computation element interworking with network management system for transparent mesh networks," Proc. IEEE/OSA OFC/NFOEC 2008, Feb. 2008, San Diego, CA, pp. 1‒10.

Tucker, R. S.

G. Shen and R. S. Tucker, "Translucent optical networks: the way forward," IEEE Commun. Mag. 45, (2), 48‒54 (2007).
[CrossRef]

Valcarenghi, L.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Varvarigos, E.

Vokkarane, V. M.

Vu, H. L.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, "Performance analyses of optical burst-switching networks," IEEE J. Sel. Areas Commun. 21, (7), 1187‒1197 (2003).
[CrossRef]

Wang, B.

Y. Fan and B. Wang, "Physical impairment aware scheduling in optical burst switched networks," Photonic Network Commun. 18, (2), 244‒254 (2009).
[CrossRef]

Wang, C.

W. Zhang, J. Tang, K. E. Nygard, and C. Wang, "REPARE: Regenerator placement and routing establishment in translucent networks," Proc. IEEE Globecom 2009, Nov. 2009, pp. 1‒7.

Wei, J. Y.

Wessäly, R.

S. Orlowski, M. Pióro, A. Tomaszewski, and R. Wessäly, "SNDlib 1.0 survivable network design library," Networks 55, (3), 276‒286 (2010).

White, J.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, "Performance analyses of optical burst-switching networks," IEEE J. Sel. Areas Commun. 21, (7), 1187‒1197 (2003).
[CrossRef]

Wosinska, L.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

Yang, X.

X. Yang and B. Ramamurthy, "Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing," Photonic Network Commun. 10, (1), 39‒53 (2005).
[CrossRef]

Yoo, M.

C. Qiao and M. Yoo, "Optical burst switching (OBS)–A new paradigm for an optical Internet," J. High Speed Netw. 8, (1), 69‒84 (1999).

Zhang, W.

W. Zhang, J. Tang, K. E. Nygard, and C. Wang, "REPARE: Regenerator placement and routing establishment in translucent networks," Proc. IEEE Globecom 2009, Nov. 2009, pp. 1‒7.

Zukerman, M.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, "Performance analyses of optical burst-switching networks," IEEE J. Sel. Areas Commun. 21, (7), 1187‒1197 (2003).
[CrossRef]

Comput. Netw.

S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Solé-Pareta, and I. Tomkos, "A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks," Comput. Netw. 53, (7), 926‒944 (2009).
[CrossRef]

IEEE Commun. Mag.

R. Martínez, C. Pinart, F. Cugini, N. Andriolli, L. Valcarenghi, P. Castoldi, L. Wosinska, J. Comellas, and G. Junyent, "Challenges and requirements for introducing impairment-awareness into the management and control planes of ASON/GMPLS WDM networks," IEEE Commun. Mag. 44, (12), 76‒85 (2006).
[CrossRef]

G. Shen and R. S. Tucker, "Translucent optical networks: the way forward," IEEE Commun. Mag. 45, (2), 48‒54 (2007).
[CrossRef]

R. Muñoz, R. Martínez, and R. Casellas, "Challenges for GMPLS lightpath provisioning in transparent optical networks: wavelength constraints in routing and signalling," IEEE Commun. Mag. 47, (8), 26‒34 (2009).

IEEE J. Sel. Areas Commun.

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, "Performance analyses of optical burst-switching networks," IEEE J. Sel. Areas Commun. 21, (7), 1187‒1197 (2003).
[CrossRef]

Int. Trans. Oper. Res.

R. Reis, M. Ritt, L. Buriol, and M. Resende, "A biased random-key genetic algorithm for OSPF and DEFT routing to minimize network congestion," Int. Trans. Oper. Res. 18, (3), 401‒423 (2011).

J. Global. Optim.

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global. Optim. 50, (3), 503‒518 (2011).
[CrossRef]

J. Heuristics

J. Gonçalves and M. Resende, "Biased random-key genetic algorithms for combinatorial optimization," J. Heuristics (posted 27 August 2010, in press).

J. High Speed Netw.

C. Qiao and M. Yoo, "Optical burst switching (OBS)–A new paradigm for an optical Internet," J. High Speed Netw. 8, (1), 69‒84 (1999).

J. Lightwave Technol.

J. Netw.

O. Pedrola, S. Rumley, M. Klinkowski, D. Careglio, C. Gaumier, and J. Solé-Pareta, "JAVOBS: a flexible simulator for OBS network architectures," J. Netw. 5, (2), 256‒264 (2010).

J. Opt. Commun. Netw.

J. Sel. Areas Commun.

J. Triay and C. Cervelló-Pastor, "An ant-based algorithm for distributed routing and wavelength assignment in dynamic optical networks," J. Sel. Areas Commun. 28, (4), 542‒552 (2010).
[CrossRef]

Lect. Notes Comput. Sci.

R. Garlick and R. Barr, "Dynamic wavelength routing in WDM networks via ant colony optimization," Lect. Notes Comput. Sci. 2463, 27‒41 (2002).

Networks

S. Orlowski, M. Pióro, A. Tomaszewski, and R. Wessäly, "SNDlib 1.0 survivable network design library," Networks 55, (3), 276‒286 (2010).

Opt. Switching Netw.

M. Klinkowski, J. Pedro, D. Careglio, M. Pióro, J. Pires, P. Monteiro, and J. Solé-Pareta, "An overview of routing methods in optical burst switching networks," Opt. Switching Netw. 7, (2), 41‒53 (2010).
[CrossRef]

Photonic Network Commun.

S. Ngo, X. Jiang, and S. Horiguchi, "An ant-based approach for dynamic RWA in optical WDM networks," Photonic Network Commun. 11, (1), 39‒48 (2006).
[CrossRef]

H. Pereira, D. A. R. Chaves, C. J. A. Bastos-Filho, and J. F. Martins-Filho, "OSNR model to consider physical layer impairments in transparent optical networks," Photonic Network Commun. 18, (2), 137‒149 (2009).
[CrossRef]

S. De Maesschalck, D. Colle, I. Lievens, M. Pickavet, P. Demeester, C. Mauz, M. Jaeger, R. Inkret, B. Mikac, and J. Derkacz, "Pan-European optical transport networks: An availability-based comparison," Photonic Network Commun. 5, (3), 203‒225 (2003).
[CrossRef]

Y. Fan and B. Wang, "Physical impairment aware scheduling in optical burst switched networks," Photonic Network Commun. 18, (2), 244‒254 (2009).
[CrossRef]

X. Yang and B. Ramamurthy, "Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing," Photonic Network Commun. 10, (1), 39‒53 (2005).
[CrossRef]

Theor. Comput. Sci.

M. Dorigo and C. Blum, "Ant colony optimization theory: A survey," Theor. Comput. Sci. 344, (2–3), 243‒278 (2005).
[CrossRef]

Other

IBM ILOG CPLEX [Online]. Available: http://www-01.ibm.com/software/integration/optimization/cplex/

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Figures (7)

Fig. 1
Fig. 1

T-OBS node architecture [8].

Fig. 2
Fig. 2

Signal path between two TAS OBS core nodes.

Fig. 3
Fig. 3

(Color online) OSNR evaluation for some European and American network end-to-end optical paths.

Fig. 4
Fig. 4

(Color online) RPD algorithms performance in the USA–Can, German, Core and Large networks under different B QoT targets.

Fig. 5
Fig. 5

(Color online) BRKGA versus KLS performance comparison in the German network.

Fig. 6
Fig. 6

(Color online) BRKGA versus RG performance comparison in the Large network.

Fig. 7
Fig. 7

The topologies considered: (a) Large (37 nodes), (b) Basic (28 nodes), (c) Core (16 nodes), (d) USA–Can (39 nodes), (e) German (50 nodes).

Tables (10)

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Table I Parameter Values Considered

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Table II P o Size Values

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Table III BRKGA Parameter Values

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Table IV ACO Parameter Values Evaluated

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Table V ACO Parameters Selected for Each Network Topology

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Table VI MP1 Results and Optimality Gaps Obtained by CPLEX

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Table VII RPD Algorithms Results Evaluation

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Table VIII RPD Algorithms Execution Time (Seconds)

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Table IX Share of Burst Losses for a BRKGA Dimensioning in the Large Network

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Table X Path Characteristics: Number of Nodes Traversed and Distance (km)

Equations (31)

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T osnr = T osnr−min + T osnr−pen ,
P osnr = 1 / i = 1 k 1 L osnr i + i = 1 k 1 N osnr i ,
L osnr i = 1 / j = 1 r 1 A S osnr j ,
A S osnr j [ dB ] = P j [ dBm ] Q N [ dBm ] F j [ dB ] G j [ dB ] ,
F eq = F wc−soa + M F sw−soa 1 G wc−soa L splitter + F edfa−booster 1 G wc−soa G sw−soa L splitter L combiner ,
G eq = G wc−soa G sw−soa G edfa−booster L splitter L combiner .
minimize y
subject to p P e x p h d y 0 , e E ,
p P d x p = 1 , d D ,
x p { 0 , 1 } ,    p P ,
minimize e E p P e x p h d
subject to p P e x p h d y , e E
s S p z p s = 1 , p P o ,
z p s { 0 , 1 } , s S p , p P o .
δ = max { | N p | : p P o } .
Θ = 2 ( δ 2 ) 1 .
O ( M | E | | P o | Θ ) ,
O P o ( δ 1 ) ( δ 2 ) 2 ,
τ i j = v s j φ v .
η ( c i j ) = 1 v s j σ ,
c i j = max c i j C i { τ i j [ η ( c i j ) ] β } if  r r 0 Q if  r > r 0 ,
p ( c i j ) = τ i j [ η ( c i j ) ] β c i k C i τ i k [ η ( c i k ) ] β .
φ v ( t + 1 ) = φ v ( t ) + α e ψ Δ r , v s j ,
φ v ( t + 1 ) = ( 1 ϑ ) φ ( t ) + ϑ e ϕ ( g ( R x ) g ( R B E S T ) ) , v s j : Ω x s j , Ω x Ω I T ,
O ( GlobItr MaxAnts | P o | Θ ) ,
O ( M G | N | P r o c . 4 ) .
P r o c . 4 = O ( | P o | Θ ( δ 2 ) ) ,
ρ v o = p P o : V p v s S p : s v z p s ρ p .
ρ v = p P o : V p v ρ p
F v ( ρ v o ) = B 1 ( ρ v o , B QoT ) ,
B ( ρ v o , r ) = ( ρ v o ) r / r ! k = 0 r ( ρ v o ) k / k ! .