Abstract

Network operators are facing the problem of dimensioning their networks for the expected huge IP traffic volumes while keeping constant or even reducing the connectivity prices. Therefore, new architectural solutions able to cope with the expected traffic increase in a more cost-effective way are needed. In this work, we study the survivable IP/multi-protocol label switching (MPLS) over wavelength switched optical network (WSON) multilayer network problem as a capital expenditure (CAPEX) minimization problem. Two network approaches providing survivability against optical links, IP/MPLS nodes, and opto-electronic port failures are compared: the classical overlay approach where two redundant IP/MPLS networks are deployed, and the new joint multilayer approach which provides the requested survivability through an orchestrated interlayer recovery scheme which minimizes the over-dimensioning of IP/MPLS nodes. Mathematical programming models are developed for both approaches. Solving these models, however, becomes impractical for realistic networks. In view of this, evolutionary heuristics based on the biased random-key genetic algorithm framework are also proposed. Exhaustive experiments on several reference network scenarios illustrate the effectiveness of the proposed approach in minimizing network CAPEX.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Chiu and J. Strand, "Joint IP/optical layer restoration after a router failure," Proc. OFC 2001, Mar. 2001, Anaheim, CA, USA.
  2. P. Chołda and A. Jajszczyk, "Recovery and its quality in multilayer networks," J. Lightwave Technol. 28, 372‒389 (2010).
    [CrossRef]
  3. C. Chigan, G. W. Atkinson, and R. Nagarajan, "Cost effectiveness of joint multilayer protection in packet-over-optical networks," J. Lightwave Technol. 21, 2694‒2704 (2003).
    [CrossRef]
  4. L. Velasco, F. Agraz, R. Martínez, R. Casellas, S. Spadaro, R. Muñoz, and G. Junyent, "GMPLS-based multi-domain restoration: Analysis, strategies, policies and experimental assessment," J. Opt. Commun. Netw. 2, 427‒441 (2010).
    [CrossRef]
  5. K. Zhu and B. Mukherjee, "Traffic grooming in an optical WDM mesh network," IEEE J. Sel. Areas Commun. 20, 122‒133 (2002).
    [CrossRef]
  6. B. Chen, G. Rouskas, and R. Dutta, "On hierarchical traffic grooming in WDM networks," IEEE/ACM Trans. Netw. 16, 1226‒1238 (2008).
    [CrossRef]
  7. B. Chen, G. Rouskas, and R. Dutta, "Clustering methods for hierarchical traffic grooming in large-scale mesh WDM networks," J. Opt. Commun. Netw. 2, 502‒514 (2010).
    [CrossRef]
  8. X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
    [CrossRef]
  9. J. Gonçalves and M. Resende, "Biased random-key genetic algorithms for combinatorial optimization," J. Heuristics posted Aug. 2010, in press.
  10. R. Morais, C. Pavan, A. Pinto, and C. Requejo, "Genetic algorithm for the topological design of survivable optical transport networks," J. Opt. Commun. Netw. 3, 17‒26 (2011).
    [CrossRef]
  11. I. de Miguel, R. Vallejos, A. Beghelli, and R. Durán, "Genetic algorithm for joint routing and dimensioning of dynamic WDM networks," J. Opt. Commun. Netw. 1, 608‒621 (2009).
    [CrossRef]
  12. "Generalized multi-protocol label switching (GMPLS) architecture," IETF RFC-3945, E. Mannie, ed., Oct. 2004, [Online]. Available: http://www.ietf.org/rfc/rfc3945.txtPlease check whether Ref. [12] given is ok..
  13. W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
    [CrossRef]
  14. G. Chiruvolu, A. Ge, D. Elie-Dit-Cosaque, M. Ali, and J. Rouyer, "Issues and approaches on extending Ethernet beyond LANs," IEEE Commun. Mag. 42, 80‒86 (2004).
    [CrossRef]
  15. CPLEX [Online]. Available: http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/
  16. T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global Optim. 50, 503‒518 (2010).
    [CrossRef]
  17. 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, 401‒423 (2011).
  18. J. Gonçalves and M. Resende, "A parallel multi population genetic algorithm for a constrained two dimensional orthogonal packing problem," J. Comb. Optim. 22, 180‒201 (2010).
    [CrossRef]
  19. C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed., Springer, New York, NY, USA, 2004.
  20. R. Huelsermann, M. Gunkel, C. Meusburger, and D. Schupke, "Cost modeling and evaluation of capital expenditures in optical multilayer networks," J. Opt. Netw. 7, 814‒833 (2008).
    [CrossRef]

2011 (2)

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, 401‒423 (2011).

R. Morais, C. Pavan, A. Pinto, and C. Requejo, "Genetic algorithm for the topological design of survivable optical transport networks," J. Opt. Commun. Netw. 3, 17‒26 (2011).
[CrossRef]

2010 (7)

P. Chołda and A. Jajszczyk, "Recovery and its quality in multilayer networks," J. Lightwave Technol. 28, 372‒389 (2010).
[CrossRef]

L. Velasco, F. Agraz, R. Martínez, R. Casellas, S. Spadaro, R. Muñoz, and G. Junyent, "GMPLS-based multi-domain restoration: Analysis, strategies, policies and experimental assessment," J. Opt. Commun. Netw. 2, 427‒441 (2010).
[CrossRef]

B. Chen, G. Rouskas, and R. Dutta, "Clustering methods for hierarchical traffic grooming in large-scale mesh WDM networks," J. Opt. Commun. Netw. 2, 502‒514 (2010).
[CrossRef]

J. Gonçalves and M. Resende, "A parallel multi population genetic algorithm for a constrained two dimensional orthogonal packing problem," J. Comb. Optim. 22, 180‒201 (2010).
[CrossRef]

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global Optim. 50, 503‒518 (2010).
[CrossRef]

X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
[CrossRef]

W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
[CrossRef]

2009 (1)

2008 (2)

2004 (1)

G. Chiruvolu, A. Ge, D. Elie-Dit-Cosaque, M. Ali, and J. Rouyer, "Issues and approaches on extending Ethernet beyond LANs," IEEE Commun. Mag. 42, 80‒86 (2004).
[CrossRef]

2003 (1)

2002 (1)

K. Zhu and B. Mukherjee, "Traffic grooming in an optical WDM mesh network," IEEE J. Sel. Areas Commun. 20, 122‒133 (2002).
[CrossRef]

Agraz, F.

Ali, M.

G. Chiruvolu, A. Ge, D. Elie-Dit-Cosaque, M. Ali, and J. Rouyer, "Issues and approaches on extending Ethernet beyond LANs," IEEE Commun. Mag. 42, 80‒86 (2004).
[CrossRef]

Atkinson, G. W.

Beghelli, A.

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, 401‒423 (2011).

Casella, G.

C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed., Springer, New York, NY, USA, 2004.

Casellas, R.

Chen, B.

B. Chen, G. Rouskas, and R. Dutta, "Clustering methods for hierarchical traffic grooming in large-scale mesh WDM networks," J. Opt. Commun. Netw. 2, 502‒514 (2010).
[CrossRef]

B. Chen, G. Rouskas, and R. Dutta, "On hierarchical traffic grooming in WDM networks," IEEE/ACM Trans. Netw. 16, 1226‒1238 (2008).
[CrossRef]

Chigan, C.

Chiruvolu, G.

G. Chiruvolu, A. Ge, D. Elie-Dit-Cosaque, M. Ali, and J. Rouyer, "Issues and approaches on extending Ethernet beyond LANs," IEEE Commun. Mag. 42, 80‒86 (2004).
[CrossRef]

Chiu, A.

A. Chiu and J. Strand, "Joint IP/optical layer restoration after a router failure," Proc. OFC 2001, Mar. 2001, Anaheim, CA, USA.

Cholda, P.

Colitti, W.

W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
[CrossRef]

Colle, D.

W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
[CrossRef]

de Miguel, I.

Durán, R.

Dutta, R.

B. Chen, G. Rouskas, and R. Dutta, "Clustering methods for hierarchical traffic grooming in large-scale mesh WDM networks," J. Opt. Commun. Netw. 2, 502‒514 (2010).
[CrossRef]

B. Chen, G. Rouskas, and R. Dutta, "On hierarchical traffic grooming in WDM networks," IEEE/ACM Trans. Netw. 16, 1226‒1238 (2008).
[CrossRef]

Elie-Dit-Cosaque, D.

G. Chiruvolu, A. Ge, D. Elie-Dit-Cosaque, M. Ali, and J. Rouyer, "Issues and approaches on extending Ethernet beyond LANs," IEEE Commun. Mag. 42, 80‒86 (2004).
[CrossRef]

Ge, A.

G. Chiruvolu, A. Ge, D. Elie-Dit-Cosaque, M. Ali, and J. Rouyer, "Issues and approaches on extending Ethernet beyond LANs," IEEE Commun. Mag. 42, 80‒86 (2004).
[CrossRef]

Gonçalves, J.

J. Gonçalves and M. Resende, "A parallel multi population genetic algorithm for a constrained two dimensional orthogonal packing problem," J. Comb. Optim. 22, 180‒201 (2010).
[CrossRef]

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

Gunkel, M.

Huelsermann, R.

Jajszczyk, A.

Junyent, G.

Lemeire, J.

W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
[CrossRef]

Li, L.

X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
[CrossRef]

Luo, H.

X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
[CrossRef]

Martínez, R.

Meusburger, C.

Morais, R.

Mukherjee, B.

K. Zhu and B. Mukherjee, "Traffic grooming in an optical WDM mesh network," IEEE J. Sel. Areas Commun. 20, 122‒133 (2002).
[CrossRef]

Muñoz, R.

Nagarajan, R.

Noronha, T.

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global Optim. 50, 503‒518 (2010).
[CrossRef]

Nowé, A.

W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
[CrossRef]

Pavan, C.

Pickavet, M.

W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
[CrossRef]

Pinto, A.

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, 401‒423 (2011).

Requejo, C.

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, 401‒423 (2011).

J. Gonçalves and M. Resende, "A parallel multi population genetic algorithm for a constrained two dimensional orthogonal packing problem," J. Comb. Optim. 22, 180‒201 (2010).
[CrossRef]

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global Optim. 50, 503‒518 (2010).
[CrossRef]

J. Gonçalves and M. Resende, "Biased random-key genetic algorithms for combinatorial optimization," J. Heuristics posted Aug. 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, 503‒518 (2010).
[CrossRef]

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, 401‒423 (2011).

Robert, C. P.

C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed., Springer, New York, NY, USA, 2004.

Rouskas, G.

B. Chen, G. Rouskas, and R. Dutta, "Clustering methods for hierarchical traffic grooming in large-scale mesh WDM networks," J. Opt. Commun. Netw. 2, 502‒514 (2010).
[CrossRef]

B. Chen, G. Rouskas, and R. Dutta, "On hierarchical traffic grooming in WDM networks," IEEE/ACM Trans. Netw. 16, 1226‒1238 (2008).
[CrossRef]

Rouyer, J.

G. Chiruvolu, A. Ge, D. Elie-Dit-Cosaque, M. Ali, and J. Rouyer, "Issues and approaches on extending Ethernet beyond LANs," IEEE Commun. Mag. 42, 80‒86 (2004).
[CrossRef]

Schupke, D.

Shen, F.

X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
[CrossRef]

Spadaro, S.

Steenhaut, K.

W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
[CrossRef]

Strand, J.

A. Chiu and J. Strand, "Joint IP/optical layer restoration after a router failure," Proc. OFC 2001, Mar. 2001, Anaheim, CA, USA.

Vallejos, R.

Velasco, L.

Wang, L.

X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
[CrossRef]

Wang, S.

X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
[CrossRef]

Zhang, X.

X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
[CrossRef]

Zhu, K.

K. Zhu and B. Mukherjee, "Traffic grooming in an optical WDM mesh network," IEEE J. Sel. Areas Commun. 20, 122‒133 (2002).
[CrossRef]

IEEE Commun. Mag. (1)

G. Chiruvolu, A. Ge, D. Elie-Dit-Cosaque, M. Ali, and J. Rouyer, "Issues and approaches on extending Ethernet beyond LANs," IEEE Commun. Mag. 42, 80‒86 (2004).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

K. Zhu and B. Mukherjee, "Traffic grooming in an optical WDM mesh network," IEEE J. Sel. Areas Commun. 20, 122‒133 (2002).
[CrossRef]

IEEE/ACM Trans. Netw. (1)

B. Chen, G. Rouskas, and R. Dutta, "On hierarchical traffic grooming in WDM networks," IEEE/ACM Trans. Netw. 16, 1226‒1238 (2008).
[CrossRef]

Int. Trans. Oper. Res. (1)

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, 401‒423 (2011).

J. Comb. Optim. (1)

J. Gonçalves and M. Resende, "A parallel multi population genetic algorithm for a constrained two dimensional orthogonal packing problem," J. Comb. Optim. 22, 180‒201 (2010).
[CrossRef]

J. Global Optim. (1)

T. Noronha, M. Resende, and C. Ribeiro, "A biased random-key genetic algorithm for routing and wavelength assignment," J. Global Optim. 50, 503‒518 (2010).
[CrossRef]

J. Heuristics (1)

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

J. Lightwave Technol. (2)

J. Opt. Commun. Netw. (4)

J. Opt. Netw. (1)

Photon. Netw. Commun. (1)

X. Zhang, F. Shen, L. Wang, S. Wang, L. Li, and H. Luo, "Two-layer mesh network optimization based on inter-layer decomposition," Photon. Netw. Commun. 21, 310‒320 (2010).
[CrossRef]

Photonic Network Commun. (1)

W. Colitti, K. Steenhaut, D. Colle, M. Pickavet, J. Lemeire, and A. Nowé, "Integrated routing in GMPLS-based IP/WDM networks," Photonic Network Commun. 21, 238‒252 (2010).
[CrossRef]

Other (4)

A. Chiu and J. Strand, "Joint IP/optical layer restoration after a router failure," Proc. OFC 2001, Mar. 2001, Anaheim, CA, USA.

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

"Generalized multi-protocol label switching (GMPLS) architecture," IETF RFC-3945, E. Mannie, ed., Oct. 2004, [Online]. Available: http://www.ietf.org/rfc/rfc3945.txtPlease check whether Ref. [12] given is ok..

C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed., Springer, New York, NY, USA, 2004.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(Color online) (a) The overlay network approach, (b) single link/node failure recovery.

Fig. 2
Fig. 2

(Color online) (a) Joint approach, (b) link failure recovery, and (c) node failure recovery.

Fig. 3
Fig. 3

(Color online) Sample optical network topologies used in this paper: 21-node Spanish Telefónica (left), 20-node British Telecom (center), and 21-node Deutsche Telecom (right). A table with details of the IP/MPLS topologies as well as the IP/MPLS traffic mix is also provided.

Fig. 4
Fig. 4

Average CAPEX savings for several C r e s t / C unp ratios by implementing the joint network approach as a function of the cost per km of unprotected lightpaths in the TEL (left), BT (center), and DT (right) networks.

Fig. 5
Fig. 5

Switching capacity of IP/MPLS nodes (left), relative joint-overlay used switching capacity (center), and relative joint-overlay average port bit-rate (right) as a function of the network load in the TEL (top), BT (middle), and DT (bottom) networks. Each load ( 4 Gbps 1 . 4 5 ( i 1 ) ) is identified by the exponent i.

Fig. 6
Fig. 6

Gap to the best solution against heuristics running time.

Tables (6)

Tables Icon

Table I Overlay Approach Algorithm

Tables Icon

Table II Sizes of the Models for Both Approaches

Tables Icon

Table III Decoder Algorithm for the Joint Approach

Tables Icon

Table IV BRKGA Parameter Values

Tables Icon

Table V Cost of IP/MPLS Nodes and OE Ports (c.u.)

Tables Icon

Table VI OE Ports Analysis (TEL Network; Intensity i = 4 )

Equations (38)

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

COST Equipment Joint = v V ∖V V s S ( v ) i P T m p c i + o p c i ρ i v s + j R T r c j π j v ,
COST Lightpaths Joint = C rest e E c C ( e ) k K ( e ) φ ec 0 k l L l e n l p a t h l k ,
M i n i m i z e CAPEX Joint = COST Equipment Joint + COST Lightpaths Joint
e E 1 ( v ) c C ( e ) ω dec f + h d e E 2 ( v ) c C ( e ) ω dec f = 1 d D , f F , v S D ( d ) V V ¯ ,
e E 1 ( v ) E 3 ( v ) c C ( e ) ω dec f + h d e E 4 ( v ) c C ( e ) ω dec f = 1 d D , f F , v S D ( d ) V V ,
e E 1 ( v ) c C ( e ) ω dec f + o u t d e E 3 ( v ) c C ( e ) ω dec f 2 d D , f F , v S D ( d ) ¯ V T V I ,
e E 1 ( v ) c C ( e ) ω dec f + o u t d e E 3 ( v ) c C ( e ) ω dec f 0 d D , f F , v S D ( d ) ¯ V M V V ,
e E 1 ( v ) e e c C ( e ) ω d e c f + o u t d e E 3 ( v ) e e c C ( e ) ω d e c f c C ( e ) ω dec f d D , f F , v S D ( d ) ¯ V T V I , e E 1 E 3 ,
d D ω dec f M k K ( e ) w f a i l f k φ ec f k f F , e E , c C ( e ) ,
k K ( e ) φ e c f k 1 f F , e E , c C ( e ) ,
e E c C ( e ) k K ( e ) p a t h l k φ e c f k w l f F , l L O ,
d D ω dec f M s S ( v ) m f a i l f v s ψ ec f v s f F , e E , c C ( e ) , v I ( e ) ,
s S ( v ) ψ ec f v s 1 f F , e E , c C ( e ) , v I ( e ) ,
e E ( v ) c C ( e ) ψ ec f v s 1 f F , v V , s S ,
d D b d ω dec f M 1 ψ ec f v s τ f v s v V V V , s S ( v ) , e E ( v ) , c C ( e ) , f F ,
τ f v s i P T p k i ρ i v s v V V V , s S ( v ) , f F ,
i P T ρ i v s 1 v V V V , s S ( v ) ,
s S ( v ) τ f v s j R T r k j π j v v V V V , f F ,
s S ( v ) i P T ρ i v s j R T r p k j π j v v V V V , f F ,
j R T π j v 1 v V V V ,
v I ( e ) s S ( v ) 1 m f a i l f v s ψ ec 0 v s + M 1 ω dec f γ d f d D , f F 0 , e E , c C ( e ) ,
c C ( e ) ω dec 0 c C ( e ) ω dec f 1 γ d f d D , f F , e E ,
c C ( e ) ω dec 0 c C ( e ) ω dec f γ d f 1 d D , f F , e E ,
γ d f , ω dec f , φ ec f k , ψ ec f v s , ρ i v s , π j v 0 , 1 , τ f v s Z + .
COST Equipment Overlay = 4 e E ∖E 5 c C ( e ) i P T m p c i + o p c i δ eci + v V M j R T r c j π j v + 2 v V T V I j R T r c j π j v ,
COST L i g h t p a t h s Overlay = C unp p P k K κ p k 0 + κ p k 1 l L l e n l p a t h l k .
M i n i m i z e COST Equipment Overlay + C unp e E ∖E 5 c C ( e ) i P T d i s t e δ eci
d D B d ω d e c i P T p k i δ eci e E E 5 , c C ( e ) ,
i P T δ eci 1 e E E 5 , c C ( e ) ,
e E ( v ) ∖E 5 d D B d ω d e j R T r k j π j v v V V V ,
2 e E ( v ) ∖E 5 c C ( e ) δ eci j R T r p k j π j v v V M ,
e E ( v ) ∖E 5 c C ( e ) δ eci j R T r p k j π j v v V T V I ,
δ eci { 0 , 1 } .
M i n i m i z e COST Lightpaths Overlay
k K ( p ) κ p k q = 1 p P , q 0 , 1 ,
p a t h l k κ p k 0 + κ p k 1 1 p P , k K ( p ) , l L o ,
p P k K ( p ) p a t h l k κ p k 0 + κ p k 1 w l l L o ,
κ p k q 0 , 1 .