Abstract

From the analysis of statistical properties of real networks, it is found that general extreme value (GEV) distribution provides an accurate model for link length statistics of optical transport networks (OTNs). The parameters of GEV distribution can be estimated from the average link length of the OTNs. Expressions for average link lengths, based only on the knowledge of network coverage area and number of nodes, are improved for better accuracy. It is shown that the optimized GEV distribution estimates the link statistics of OTNs with good accuracy (Kolmogorov–Smirnov statistic less than 0.18).

© 2013 Optical Society of America

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References

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  1. S. K. Korotky, “Network global expectation model: A statistical formalism for quickly quantifying network needs and costs,” J. Lightwave Technol., vol.  22, no. 3, pp. 703–722, Mar. 2004.
    [CrossRef]
  2. J. P. Cardenas, A. Santiago, J. C. Losada, R. M. Benito, and M. L. Mouronte, “On the topology of optical transport networks,” J. Phys.: Conf. Ser., vol.  246, 012013, 2010.
    [CrossRef]
  3. C. Pavan, R. M. Morais, J. R. F. da Rocha, and A. N. Pinto, “Generating realistic optical transport network topologies,” J. Opt. Commun. Netw., vol.  2, no. 1, pp. 80–90, Jan. 2010.
    [CrossRef]
  4. R. M. Morais, C. Pavan, C. Requejo, and A. N. Pinto, “Genetic algorithm for the topological design of survivable optical transport networks,” J. Opt. Commun. Netw., vol.  2, no. 1, pp. 80–90, Jan. 2010.
    [CrossRef]
  5. S. K. Korotky, R. Essiambre, and R. W. Tkach, “Expectations of optical network traffic gain afforded by bit rate adaptive transmission,” Bell Labs Tech. J., vol.  14, no. 4, pp. 285–296, Feb. 2010.
    [CrossRef]
  6. S. K. Korotky and K. N. Oikonomou, “Scaling of most-likely traffic patterns of hose- and cost-constrained ring and mesh networks,” in Proc. Optical Fiber Communications Conf. (OFC), Anaheim, CA, Mar. 2006.
  7. A. Dwivedi and R. E. Wagner, “Traffic model for USA long-distance optical network,” in Proc. Optical Fiber Communications Conf. (OFC), Baltimore, MD, 2000, pp. 156–158.
  8. A. N. Pinto, C. Pavan, and R. M. Morais, “Dimensioning optical networks: A practical approach,” in Proc. of 12th Int. Conf. Transparent Optical Networks (ICTON), Munich, Germany, Jan. 2010.
  9. J. F. Labourdette, E. Bouillet, R. Ramamurthy, and A. A. Akyamaç, “Fast approximate dimensioning and performance analysis of mesh optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 4, pp. 906–917, Aug. 2005.
    [CrossRef]
  10. M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
    [CrossRef]
  11. Reference Optical Networks, Apr. 2013 [Online]. Available: http://www.av.it.pt/anp/on/refnet2.html .
  12. The Lonestar Education And Research Network (LEARN) [Online]. Available: http://www.tx-learn.org/images/map-large.jpg .
  13. The CompuServe Backbone Optical Transport Network [Online]. Available: http://www.nthelp.com/images/compuserve.jpg .
  14. The Abilene Backbone Optical Transport Network [Online]. Available: http://www.internet2.edu/2004AR/abilene_map_large.cfm .
  15. GARR-B, The Italian Research and Education Network (NREN) [Online]. Available: http://www.garr.it/index.php/rete/garr-x#mappa .
  16. Swedish University Computer Network, OptoSUNET [Online]. Available: http://www.sunet.se/download/18.6d7c8917128274d3dd0800 06061/optosunetbroschyr_eng.pdf .
  17. The High Speed Hibernia Atlantic USA Networks [Online]. Available: http://www.hiberniaatlantic.com/US_network.html .
  18. The following reference is the source of the 31 Node Italian Backbone Optical Transport Network (IBN31): A. Eira, J. Pedro, and J. Pires, “Optimized design of shared restoration in flexible-grid transparent optical networks,” in Optical Fiber Communication Conf., Los Angeles, CA, Mar.2012, paper JTh2A.37.
  19. Bulgarian Backbone Optical Transport Network [Online]. Available: http://www.gcn.bg/bg_backbone.html .
  20. The following reference is the source of the European Core Optical Transport Network (COST37): J. Tapolcai, P. H. Ho, and A. Haque, “TROP: A novel approximate link-state dissemination framework for dynamic survivable routing in MPLS networks,” IEEE Trans. Parallel Distrib. Syst., vol.  19, no. 3, pp. 311–322, Mar. 2008.
  21. The Chinese Education and Research Network (CERNET) [Online]. Available: http://www.edu.cn/20060111/3170194.shtml .
  22. The Next Generation Core Optical Networks (CORONET) [Online]. Available: http://monarchna.com/topology.html .
  23. S. Kotz and S. Nadarajah, Extreme Value Distributions: Theory and Applications. London, UK: Imperial College Press, 2000.
  24. The World Atlas [Online]. Available: http://www.worldatlas.com/ .

2011 (1)

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

2010 (4)

J. P. Cardenas, A. Santiago, J. C. Losada, R. M. Benito, and M. L. Mouronte, “On the topology of optical transport networks,” J. Phys.: Conf. Ser., vol.  246, 012013, 2010.
[CrossRef]

C. Pavan, R. M. Morais, J. R. F. da Rocha, and A. N. Pinto, “Generating realistic optical transport network topologies,” J. Opt. Commun. Netw., vol.  2, no. 1, pp. 80–90, Jan. 2010.
[CrossRef]

R. M. Morais, C. Pavan, C. Requejo, and A. N. Pinto, “Genetic algorithm for the topological design of survivable optical transport networks,” J. Opt. Commun. Netw., vol.  2, no. 1, pp. 80–90, Jan. 2010.
[CrossRef]

S. K. Korotky, R. Essiambre, and R. W. Tkach, “Expectations of optical network traffic gain afforded by bit rate adaptive transmission,” Bell Labs Tech. J., vol.  14, no. 4, pp. 285–296, Feb. 2010.
[CrossRef]

2008 (1)

The following reference is the source of the European Core Optical Transport Network (COST37): J. Tapolcai, P. H. Ho, and A. Haque, “TROP: A novel approximate link-state dissemination framework for dynamic survivable routing in MPLS networks,” IEEE Trans. Parallel Distrib. Syst., vol.  19, no. 3, pp. 311–322, Mar. 2008.

2005 (1)

J. F. Labourdette, E. Bouillet, R. Ramamurthy, and A. A. Akyamaç, “Fast approximate dimensioning and performance analysis of mesh optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 4, pp. 906–917, Aug. 2005.
[CrossRef]

2004 (1)

Akyamaç, A. A.

J. F. Labourdette, E. Bouillet, R. Ramamurthy, and A. A. Akyamaç, “Fast approximate dimensioning and performance analysis of mesh optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 4, pp. 906–917, Aug. 2005.
[CrossRef]

Barabási, A.-L.

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

Benito, R. M.

J. P. Cardenas, A. Santiago, J. C. Losada, R. M. Benito, and M. L. Mouronte, “On the topology of optical transport networks,” J. Phys.: Conf. Ser., vol.  246, 012013, 2010.
[CrossRef]

Bouillet, E.

J. F. Labourdette, E. Bouillet, R. Ramamurthy, and A. A. Akyamaç, “Fast approximate dimensioning and performance analysis of mesh optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 4, pp. 906–917, Aug. 2005.
[CrossRef]

Cardenas, J. P.

J. P. Cardenas, A. Santiago, J. C. Losada, R. M. Benito, and M. L. Mouronte, “On the topology of optical transport networks,” J. Phys.: Conf. Ser., vol.  246, 012013, 2010.
[CrossRef]

da Rocha, J. R. F.

Dwivedi, A.

A. Dwivedi and R. E. Wagner, “Traffic model for USA long-distance optical network,” in Proc. Optical Fiber Communications Conf. (OFC), Baltimore, MD, 2000, pp. 156–158.

Eira, A.

The following reference is the source of the 31 Node Italian Backbone Optical Transport Network (IBN31): A. Eira, J. Pedro, and J. Pires, “Optimized design of shared restoration in flexible-grid transparent optical networks,” in Optical Fiber Communication Conf., Los Angeles, CA, Mar.2012, paper JTh2A.37.

Essiambre, R.

S. K. Korotky, R. Essiambre, and R. W. Tkach, “Expectations of optical network traffic gain afforded by bit rate adaptive transmission,” Bell Labs Tech. J., vol.  14, no. 4, pp. 285–296, Feb. 2010.
[CrossRef]

Haque, A.

The following reference is the source of the European Core Optical Transport Network (COST37): J. Tapolcai, P. H. Ho, and A. Haque, “TROP: A novel approximate link-state dissemination framework for dynamic survivable routing in MPLS networks,” IEEE Trans. Parallel Distrib. Syst., vol.  19, no. 3, pp. 311–322, Mar. 2008.

Ho, P. H.

The following reference is the source of the European Core Optical Transport Network (COST37): J. Tapolcai, P. H. Ho, and A. Haque, “TROP: A novel approximate link-state dissemination framework for dynamic survivable routing in MPLS networks,” IEEE Trans. Parallel Distrib. Syst., vol.  19, no. 3, pp. 311–322, Mar. 2008.

Karsai, M.

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

Kaski, K.

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

Kertész, J.

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

Kivela, M.

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

Korotky, S. K.

S. K. Korotky, R. Essiambre, and R. W. Tkach, “Expectations of optical network traffic gain afforded by bit rate adaptive transmission,” Bell Labs Tech. J., vol.  14, no. 4, pp. 285–296, Feb. 2010.
[CrossRef]

S. K. Korotky, “Network global expectation model: A statistical formalism for quickly quantifying network needs and costs,” J. Lightwave Technol., vol.  22, no. 3, pp. 703–722, Mar. 2004.
[CrossRef]

S. K. Korotky and K. N. Oikonomou, “Scaling of most-likely traffic patterns of hose- and cost-constrained ring and mesh networks,” in Proc. Optical Fiber Communications Conf. (OFC), Anaheim, CA, Mar. 2006.

Kotz, S.

S. Kotz and S. Nadarajah, Extreme Value Distributions: Theory and Applications. London, UK: Imperial College Press, 2000.

Labourdette, J. F.

J. F. Labourdette, E. Bouillet, R. Ramamurthy, and A. A. Akyamaç, “Fast approximate dimensioning and performance analysis of mesh optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 4, pp. 906–917, Aug. 2005.
[CrossRef]

Losada, J. C.

J. P. Cardenas, A. Santiago, J. C. Losada, R. M. Benito, and M. L. Mouronte, “On the topology of optical transport networks,” J. Phys.: Conf. Ser., vol.  246, 012013, 2010.
[CrossRef]

Morais, R. M.

Mouronte, M. L.

J. P. Cardenas, A. Santiago, J. C. Losada, R. M. Benito, and M. L. Mouronte, “On the topology of optical transport networks,” J. Phys.: Conf. Ser., vol.  246, 012013, 2010.
[CrossRef]

Nadarajah, S.

S. Kotz and S. Nadarajah, Extreme Value Distributions: Theory and Applications. London, UK: Imperial College Press, 2000.

Oikonomou, K. N.

S. K. Korotky and K. N. Oikonomou, “Scaling of most-likely traffic patterns of hose- and cost-constrained ring and mesh networks,” in Proc. Optical Fiber Communications Conf. (OFC), Anaheim, CA, Mar. 2006.

Pan, R. K.

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

Pavan, C.

Pedro, J.

The following reference is the source of the 31 Node Italian Backbone Optical Transport Network (IBN31): A. Eira, J. Pedro, and J. Pires, “Optimized design of shared restoration in flexible-grid transparent optical networks,” in Optical Fiber Communication Conf., Los Angeles, CA, Mar.2012, paper JTh2A.37.

Pinto, A. N.

Pires, J.

The following reference is the source of the 31 Node Italian Backbone Optical Transport Network (IBN31): A. Eira, J. Pedro, and J. Pires, “Optimized design of shared restoration in flexible-grid transparent optical networks,” in Optical Fiber Communication Conf., Los Angeles, CA, Mar.2012, paper JTh2A.37.

Ramamurthy, R.

J. F. Labourdette, E. Bouillet, R. Ramamurthy, and A. A. Akyamaç, “Fast approximate dimensioning and performance analysis of mesh optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 4, pp. 906–917, Aug. 2005.
[CrossRef]

Requejo, C.

Santiago, A.

J. P. Cardenas, A. Santiago, J. C. Losada, R. M. Benito, and M. L. Mouronte, “On the topology of optical transport networks,” J. Phys.: Conf. Ser., vol.  246, 012013, 2010.
[CrossRef]

Saramäki, J.

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

Tapolcai, J.

The following reference is the source of the European Core Optical Transport Network (COST37): J. Tapolcai, P. H. Ho, and A. Haque, “TROP: A novel approximate link-state dissemination framework for dynamic survivable routing in MPLS networks,” IEEE Trans. Parallel Distrib. Syst., vol.  19, no. 3, pp. 311–322, Mar. 2008.

Tkach, R. W.

S. K. Korotky, R. Essiambre, and R. W. Tkach, “Expectations of optical network traffic gain afforded by bit rate adaptive transmission,” Bell Labs Tech. J., vol.  14, no. 4, pp. 285–296, Feb. 2010.
[CrossRef]

Wagner, R. E.

A. Dwivedi and R. E. Wagner, “Traffic model for USA long-distance optical network,” in Proc. Optical Fiber Communications Conf. (OFC), Baltimore, MD, 2000, pp. 156–158.

Bell Labs Tech. J. (1)

S. K. Korotky, R. Essiambre, and R. W. Tkach, “Expectations of optical network traffic gain afforded by bit rate adaptive transmission,” Bell Labs Tech. J., vol.  14, no. 4, pp. 285–296, Feb. 2010.
[CrossRef]

IEEE Trans. Parallel Distrib. Syst. (1)

The following reference is the source of the European Core Optical Transport Network (COST37): J. Tapolcai, P. H. Ho, and A. Haque, “TROP: A novel approximate link-state dissemination framework for dynamic survivable routing in MPLS networks,” IEEE Trans. Parallel Distrib. Syst., vol.  19, no. 3, pp. 311–322, Mar. 2008.

IEEE/ACM Trans. Netw. (1)

J. F. Labourdette, E. Bouillet, R. Ramamurthy, and A. A. Akyamaç, “Fast approximate dimensioning and performance analysis of mesh optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 4, pp. 906–917, Aug. 2005.
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Commun. Netw. (2)

J. Phys.: Conf. Ser. (1)

J. P. Cardenas, A. Santiago, J. C. Losada, R. M. Benito, and M. L. Mouronte, “On the topology of optical transport networks,” J. Phys.: Conf. Ser., vol.  246, 012013, 2010.
[CrossRef]

Phys. Rev. E (1)

M. Karsai, M. Kivela, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, “Small but slow world: How network topology and burstiness slow down spreading,” Phys. Rev. E, vol.  83, 025102, 2011.
[CrossRef]

Other (16)

Reference Optical Networks, Apr. 2013 [Online]. Available: http://www.av.it.pt/anp/on/refnet2.html .

The Lonestar Education And Research Network (LEARN) [Online]. Available: http://www.tx-learn.org/images/map-large.jpg .

The CompuServe Backbone Optical Transport Network [Online]. Available: http://www.nthelp.com/images/compuserve.jpg .

The Abilene Backbone Optical Transport Network [Online]. Available: http://www.internet2.edu/2004AR/abilene_map_large.cfm .

GARR-B, The Italian Research and Education Network (NREN) [Online]. Available: http://www.garr.it/index.php/rete/garr-x#mappa .

Swedish University Computer Network, OptoSUNET [Online]. Available: http://www.sunet.se/download/18.6d7c8917128274d3dd0800 06061/optosunetbroschyr_eng.pdf .

The High Speed Hibernia Atlantic USA Networks [Online]. Available: http://www.hiberniaatlantic.com/US_network.html .

The following reference is the source of the 31 Node Italian Backbone Optical Transport Network (IBN31): A. Eira, J. Pedro, and J. Pires, “Optimized design of shared restoration in flexible-grid transparent optical networks,” in Optical Fiber Communication Conf., Los Angeles, CA, Mar.2012, paper JTh2A.37.

Bulgarian Backbone Optical Transport Network [Online]. Available: http://www.gcn.bg/bg_backbone.html .

The Chinese Education and Research Network (CERNET) [Online]. Available: http://www.edu.cn/20060111/3170194.shtml .

The Next Generation Core Optical Networks (CORONET) [Online]. Available: http://monarchna.com/topology.html .

S. Kotz and S. Nadarajah, Extreme Value Distributions: Theory and Applications. London, UK: Imperial College Press, 2000.

The World Atlas [Online]. Available: http://www.worldatlas.com/ .

S. K. Korotky and K. N. Oikonomou, “Scaling of most-likely traffic patterns of hose- and cost-constrained ring and mesh networks,” in Proc. Optical Fiber Communications Conf. (OFC), Anaheim, CA, Mar. 2006.

A. Dwivedi and R. E. Wagner, “Traffic model for USA long-distance optical network,” in Proc. Optical Fiber Communications Conf. (OFC), Baltimore, MD, 2000, pp. 156–158.

A. N. Pinto, C. Pavan, and R. M. Morais, “Dimensioning optical networks: A practical approach,” in Proc. of 12th Int. Conf. Transparent Optical Networks (ICTON), Munich, Germany, Jan. 2010.

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Figures (12)

Fig. 1.
Fig. 1.

Comparison of the GEV distribution with the link length histogram of the USA100 [1] network (171 links).

Fig. 2.
Fig. 2.

Plot of α of GEV versus l of 40 real networks (R2=0.9676).

Fig. 3.
Fig. 3.

Plot of β of GEV versus l of 40 real networks (R2=0.9225).

Fig. 4.
Fig. 4.

Optimized versus approximate GEV distribution of link lengths of the USA100 [1] network. This estimation uses the exact average link length l.

Fig. 5.
Fig. 5.

Standard deviation versus average link length of 40 real networks follows a linear trend (R2=0.9011).

Fig. 6.
Fig. 6.

Depiction of convex and exact areas: ABCDA and ACEGA are convex areas of ABCDEA and ABCDEFGHA, respectively.

Fig. 7.
Fig. 7.

Distribution of estimated errors with convex area (Ec(%)) for 40 networks [fitted to normal distribution (μ=0.53, σ=14.15)].

Fig. 8.
Fig. 8.

Distribution of estimated errors with exact area (Ec(%)) for 40 networks [fitted to normal distribution (μ=18.99, σ=13.67)].

Fig. 9.
Fig. 9.

Effect of multiplying factors, ke when using the exact area (ke=1.22 gives minimum error).

Fig. 10.
Fig. 10.

Effect of multiplying factors, kc when using the convex area (kc=0.97 gives minimum error).

Fig. 11.
Fig. 11.

Optimized versus estimated GEV distribution of link lengths of the USA100 [1] network (171 links). This estimation uses the estimated average link length lc.

Fig. 12.
Fig. 12.

Γ(1ξ)1ξ, (blue, solid) and its approximation 11ξ0.425, (red, dotted) with respect to ξ.

Tables (5)

Tables Icon

TABLE I Best Fitting Distributions and Their Avg. KSS, L. KSS, and H. KSS Values

Tables Icon

TABLE II Real Network Topologies and Their Attributes (l and σ Are in km)

Tables Icon

TABLE III Different Areas of the 40 Real Networks (All Areas Are in Square km)

Tables Icon

TABLE IV Estimation of Average Link Lengths From Different Areas and Their Comparisons (lc, le, lg, and l Are in km)

Tables Icon

TABLE V Comparison of Errors With Optimized Multiplying Factors in Eqs. (15) and (16) (Column 3, 4, 5, and 6). Estimation of the Parameters of the Proposed Model From the Average Link Length (Column lc, αc, βc, ξc), the KSS Values of the Networks (Column KSSGE), and Their Evaluation (Column Acceptable?)

Equations (23)

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

l=1Li=1Lli,
σ=(1Li=1L(lli)2)12.
t=1+ξ(lα)/β,F(l;α,β,ξ)=F(t;ξ)=exp{[t]1/ξ},
f(l;α,β,ξ)=f(t;β,ξ)=1β[t](1/ξ)1exp{[t]1/ξ}.
α0.6577l+8.67,
β0.441l12.37.
E(l)=αβξ+βξΓ(1ξ).
lαβ=Γ(1ξ)1ξ.
Γ(1ξ)1ξ11ξ0.425.
ξ0.0887l1.5570.5297l13.927.
σ=βξΓ(12ξ)[Γ(1ξ)]2.
σ0.64l6.9.
lxAxN1.
Ex=llxl.
lekeAeN1.
lckcAcN1.
Γ(x)=0tx1etdt.
E(l)=αβξ+βξΓ(1ξ).
lαβ=Γ(1ξ)1ξ,
Γ(1ξ)1ξ11ξ0.425.
lαβ11ξ0.425.
ξlαβ0.575lαβ+0.425=0.0887l1.5580.5297l13.927.
ξ0.08870.5297=0.167.