Performance evaluation of multi-stratum resources integration based on network function virtualization in software defined elastic data center optical interconnect

Hui Yang; Jie Zhang; Yuefeng Ji; Rui Tian; Jianrui Han; Young Lee

doi:10.1364/OE.23.031192

Performance evaluation of multi-stratum resources integration based on network function virtualization in software defined elastic data center optical interconnect

Hui Yang, Jie Zhang, Yuefeng Ji, Rui Tian, Jianrui Han, and Young Lee

Optics Express
Vol. 23,
Issue 24,
pp. 31192-31205
(2015)
•https://doi.org/10.1364/OE.23.031192

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Hui Yang, Jie Zhang, Yuefeng Ji, Rui Tian, Jianrui Han, and Young Lee, "Performance evaluation of multi-stratum resources integration based on network function virtualization in software defined elastic data center optical interconnect," Opt. Express 23, 31192-31205 (2015)

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Related Topics
Table of Contents Category
- Optical Communications
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Networks (060.4250)
- Optical communications (060.4510)

About this Article
History
- Original Manuscript: September 24, 2015
- Revised Manuscript: November 8, 2015
- Manuscript Accepted: November 12, 2015
- Published: November 20, 2015

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Open Access

Abstract

Data center interconnect with elastic optical network is a promising scenario to meet the high burstiness and high-bandwidth requirements of data center services. In our previous work, we implemented multi-stratum resilience between IP and elastic optical networks that allows to accommodate data center services. In view of this, this study extends to consider the resource integration by breaking the limit of network device, which can enhance the resource utilization. We propose a novel multi-stratum resources integration (MSRI) architecture based on network function virtualization in software defined elastic data center optical interconnect. A resource integrated mapping (RIM) scheme for MSRI is introduced in the proposed architecture. The MSRI can accommodate the data center services with resources integration when the single function or resource is relatively scarce to provision the services, and enhance globally integrated optimization of optical network and application resources. The overall feasibility and efficiency of the proposed architecture are experimentally verified on the control plane of OpenFlow-based enhanced software defined networking (eSDN) testbed. The performance of RIM scheme under heavy traffic load scenario is also quantitatively evaluated based on MSRI architecture in terms of path blocking probability, provisioning latency and resource utilization, compared with other provisioning schemes.

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References

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|
Year
|
Author
|
Publication

M. Al-Fares, A. Loukissas, and A. Vahdat, “A scalable, commodity data center network architecture,” Comput. Commun. Rev. 38(4), 63–74 (2008).
[Crossref]
C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Comm. Surv. and Tutor. 14(4), 1021–1036 (2012).
[Crossref]
H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]
O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), s12–s20 (2012).
[Crossref]
W. Shieh, “OFDM for flexible high-speed optical networks,” J. Lightwave Technol. 29(10), 1560–1577 (2011).
[Crossref]
I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
[Crossref]
R. Nejabati, S. Peng, M. Channegowda, B. Guo, and D. Simeonidou, “SDN and NFV Convergence a Technology Enabler for Abstracting and Virtualising Hardware and Control of Optical Networks (Invited),” in Proceedings of Optical Fiber Communication Conference (OFC 2015), (Optical Society of America, 2015), paper W4J.6.
M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag. 50(5), 56–65 (2012).
[Crossref]
J. Fernandez-Palacios, V. López, B. Cruz, and O. Dios, “Elastic Optical Networking: An Operators Perspective,” in Proceedings of European Conference and Exhibition on Optical Communications (ECOC 2014), (Optical Society of America, 2014), paper Mo.4.2.1.
T. Tanaka, A. Hirano, and M. Jinno, “Advantages of IP over elastic optical networks using multi-flow transponders from cost and equipment count aspects,” Opt. Express 22(1), 62–70 (2014).
[Crossref] [PubMed]
R. Vilalta, R. Muñoz, R. Casellas, R. Martínez, V. López, and D. López, “Transport PCE Network Function Virtualization,” in Proceedings of European Conference and Exhibition on Optical Communications (ECOC 2014), (Optical Society of America, 2014), paper We.3.2.2.
H. Yang, J. Zhang, Y. Ji, Y. Tan, Y. Lin, J. Han, and Y. Lee, “Performance evaluation of data center service localization based on virtual resource migration in software defined elastic optical network,” Opt. Express 23(18), 23059–23071 (2015).
[Crossref] [PubMed]
R. Martínez, R. Casellas, R. Vilalta, and R. Muñoz, “Experimental assessment of GMPLS/PCE-controlled multi-flow optical transponders in flexgrid networks,” in Proceedings of Optical Fiber Communication Conference (OFC 2015), (Optical Society of America, 2015), paper Tu2B.4.
[Crossref]
L. Liu, W. R. Peng, R. Casellas, T. Tsuritani, I. Morita, R. Martínez, R. Muñoz, and S. J. B. Yoo, “Design and performance evaluation of an OpenFlow-based control plane for software-defined elastic optical networks with direct-detection optical OFDM (DDO-OFDM) transmission,” Opt. Express 22(1), 30–40 (2014).
[Crossref] [PubMed]
M. Channegowda, R. Nejabati, M. Rashidi Fard, S. Peng, N. Amaya, G. Zervas, D. Simeonidou, R. Vilalta, R. Casellas, R. Martínez, R. Muñoz, L. Liu, T. Tsuritani, I. Morita, A. Autenrieth, J. P. Elbers, P. Kostecki, and P. Kaczmarek, “Experimental demonstration of an OpenFlow based software-defined optical network employing packet, fixed and flexible DWDM grid technologies on an international multi-domain testbed,” Opt. Express 21(5), 5487–5498 (2013).
[Crossref] [PubMed]
S. Das, G. Parulkar, and N. McKeown, “Why OpenFlow/SDN can succeed where GMPLS failed,” in Proceedings of European Conference on Optical Communication (ECOC 2012), (Optical Society of America, 2012), paper Tu.1.D.1.
[Crossref]
F. Paolucci, F. Cugini, N. Hussain, F. Fresi, and L. Poti, “OpenFlow-based flexible optical networks with enhanced monitoring functionalities,” in Proceedings of European Conference and Exhibition on Optical Communications (ECOC 2012), (Optical Society of America, 2012), paper Tu.1.D.5.
[Crossref]
H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Wu, Y. Lin, J. Han, and Y. Lee, “Performance evaluation of multi-stratum resources integrated resilience for software defined inter-data center interconnect,” Opt. Express 23(10), 13384–13398 (2015).
[Crossref] [PubMed]
H. Yang, J. Zhang, Y. Zhao, Y. Ji, H. Li, Y. Lin, G. Li, J. Han, Y. Lee, and T. Ma, “Performance evaluation of time-aware enhanced software defined networking (TeSDN) for elastic data center optical interconnection,” Opt. Express 22(15), 17630–17643 (2014).
[Crossref] [PubMed]
C. Papagianni, A. Leivadeas, S. Papavassiliou, V. Maglaris, C. Cervello-Pastor, and A. Monje, “On the optimal allocation of virtual resources in cloud computing networks,” IEEE Trans. Comput. 62(6), 1060–1071 (2013).
[Crossref]
C. Wang, W. Hung, and C. Yang, “A prediction based energy conserving resources allocation scheme for cloud computing,” in Proceedings of IEEE International Conference on Granular Computing (GrC 2014), (IEEE, 2014), paper 320–324.
[Crossref]
A. Beloglazov, J. Abawajy, and R. Buyya, “Energy-aware resource allocation heuristics for efficient management of data centers for cloud computing,” Future Gener. Comput. Syst. 28(5), 755–768 (2012).
[Crossref]

2015 (3)

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

H. Yang, J. Zhang, Y. Ji, Y. Tan, Y. Lin, J. Han, and Y. Lee, “Performance evaluation of data center service localization based on virtual resource migration in software defined elastic optical network,” Opt. Express 23(18), 23059–23071 (2015).
[Crossref] [PubMed]

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Wu, Y. Lin, J. Han, and Y. Lee, “Performance evaluation of multi-stratum resources integrated resilience for software defined inter-data center interconnect,” Opt. Express 23(10), 13384–13398 (2015).
[Crossref] [PubMed]

2014 (4)

H. Yang, J. Zhang, Y. Zhao, Y. Ji, H. Li, Y. Lin, G. Li, J. Han, Y. Lee, and T. Ma, “Performance evaluation of time-aware enhanced software defined networking (TeSDN) for elastic data center optical interconnection,” Opt. Express 22(15), 17630–17643 (2014).
[Crossref] [PubMed]

T. Tanaka, A. Hirano, and M. Jinno, “Advantages of IP over elastic optical networks using multi-flow transponders from cost and equipment count aspects,” Opt. Express 22(1), 62–70 (2014).
[Crossref] [PubMed]

L. Liu, W. R. Peng, R. Casellas, T. Tsuritani, I. Morita, R. Martínez, R. Muñoz, and S. J. B. Yoo, “Design and performance evaluation of an OpenFlow-based control plane for software-defined elastic optical networks with direct-detection optical OFDM (DDO-OFDM) transmission,” Opt. Express 22(1), 30–40 (2014).
[Crossref] [PubMed]

I. Tomkos, S. Azodolmolky, J. Sole-Pareta, D. Careglio, and E. Palkopoulou, “A tutorial on the flexible optical networking paradigm: state of the art, trends, and research challenges,” Proc. IEEE 102(9), 1317–1337 (2014).
[Crossref]

2013 (2)

M. Channegowda, R. Nejabati, M. Rashidi Fard, S. Peng, N. Amaya, G. Zervas, D. Simeonidou, R. Vilalta, R. Casellas, R. Martínez, R. Muñoz, L. Liu, T. Tsuritani, I. Morita, A. Autenrieth, J. P. Elbers, P. Kostecki, and P. Kaczmarek, “Experimental demonstration of an OpenFlow based software-defined optical network employing packet, fixed and flexible DWDM grid technologies on an international multi-domain testbed,” Opt. Express 21(5), 5487–5498 (2013).
[Crossref] [PubMed]

C. Papagianni, A. Leivadeas, S. Papavassiliou, V. Maglaris, C. Cervello-Pastor, and A. Monje, “On the optimal allocation of virtual resources in cloud computing networks,” IEEE Trans. Comput. 62(6), 1060–1071 (2013).
[Crossref]

2012 (4)

A. Beloglazov, J. Abawajy, and R. Buyya, “Energy-aware resource allocation heuristics for efficient management of data centers for cloud computing,” Future Gener. Comput. Syst. 28(5), 755–768 (2012).
[Crossref]

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag. 50(5), 56–65 (2012).
[Crossref]

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), s12–s20 (2012).
[Crossref]

C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Comm. Surv. and Tutor. 14(4), 1021–1036 (2012).
[Crossref]

2011 (1)

W. Shieh, “OFDM for flexible high-speed optical networks,” J. Lightwave Technol. 29(10), 1560–1577 (2011).
[Crossref]

2008 (1)

M. Al-Fares, A. Loukissas, and A. Vahdat, “A scalable, commodity data center network architecture,” Comput. Commun. Rev. 38(4), 63–74 (2008).
[Crossref]

Abawajy, J.

Al-Fares, M.

M. Al-Fares, A. Loukissas, and A. Vahdat, “A scalable, commodity data center network architecture,” Comput. Commun. Rev. 38(4), 63–74 (2008).
[Crossref]

Amaya, N.

Autenrieth, A.

Azodolmolky, S.

Beloglazov, A.

Buyya, R.

Careglio, D.

Casellas, R.

Cervello-Pastor, C.

Channegowda, M.

Elbers, J. P.

Gerstel, O.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), s12–s20 (2012).
[Crossref]

Han, J.

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

Hirano, A.

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag. 50(5), 56–65 (2012).
[Crossref]

Ji, Y.

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

Jinno, M.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), s12–s20 (2012).
[Crossref]

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag. 50(5), 56–65 (2012).
[Crossref]

Kachris, C.

C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Comm. Surv. and Tutor. 14(4), 1021–1036 (2012).
[Crossref]

Kaczmarek, P.

Kostecki, P.

Lee, Y.

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

Leivadeas, A.

Li, G.

Li, H.

Lin, Y.

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

Liu, L.

Lord, A.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), s12–s20 (2012).
[Crossref]

Loukissas, A.

M. Al-Fares, A. Loukissas, and A. Vahdat, “A scalable, commodity data center network architecture,” Comput. Commun. Rev. 38(4), 63–74 (2008).
[Crossref]

Ma, T.

Maglaris, V.

Martínez, R.

Monje, A.

Morita, I.

Muñoz, R.

Nejabati, R.

Palkopoulou, E.

Papagianni, C.

Papavassiliou, S.

Peng, S.

Peng, W. R.

Rashidi Fard, M.

Shieh, W.

W. Shieh, “OFDM for flexible high-speed optical networks,” J. Lightwave Technol. 29(10), 1560–1577 (2011).
[Crossref]

Simeonidou, D.

Sole-Pareta, J.

Sone, Y.

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag. 50(5), 56–65 (2012).
[Crossref]

Takara, H.

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag. 50(5), 56–65 (2012).
[Crossref]

Tan, Y.

Tanaka, T.

Tomkos, I.

C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Comm. Surv. and Tutor. 14(4), 1021–1036 (2012).
[Crossref]

Tsuritani, T.

Vahdat, A.

M. Al-Fares, A. Loukissas, and A. Vahdat, “A scalable, commodity data center network architecture,” Comput. Commun. Rev. 38(4), 63–74 (2008).
[Crossref]

Vilalta, R.

Wu, J.

Yang, H.

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

Yonenaga, K.

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag. 50(5), 56–65 (2012).
[Crossref]

Yoo, S. J. B.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), s12–s20 (2012).
[Crossref]

Zervas, G.

Zhang, J.

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

Zhao, Y.

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

Comput. Commun. Rev. (1)

M. Al-Fares, A. Loukissas, and A. Vahdat, “A scalable, commodity data center network architecture,” Comput. Commun. Rev. 38(4), 63–74 (2008).
[Crossref]

Future Gener. Comput. Syst. (1)

IEEE Comm. Surv. and Tutor. (1)

C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Comm. Surv. and Tutor. 14(4), 1021–1036 (2012).
[Crossref]

IEEE Commun. Mag. (3)

H. Yang, J. Zhang, Y. Zhao, Y. Ji, J. Han, Y. Lin, and Y. Lee, “CSO: Cross Stratum Optimization for Optical as a Service,” IEEE Commun. Mag. 53(8), 130–139 (2015).
[Crossref]

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50(2), s12–s20 (2012).
[Crossref]

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag. 50(5), 56–65 (2012).
[Crossref]

IEEE Trans. Comput. (1)

J. Lightwave Technol. (1)

W. Shieh, “OFDM for flexible high-speed optical networks,” J. Lightwave Technol. 29(10), 1560–1577 (2011).
[Crossref]

Opt. Express (6)

Proc. IEEE (1)

Other (7)

R. Nejabati, S. Peng, M. Channegowda, B. Guo, and D. Simeonidou, “SDN and NFV Convergence a Technology Enabler for Abstracting and Virtualising Hardware and Control of Optical Networks (Invited),” in Proceedings of Optical Fiber Communication Conference (OFC 2015), (Optical Society of America, 2015), paper W4J.6.

R. Vilalta, R. Muñoz, R. Casellas, R. Martínez, V. López, and D. López, “Transport PCE Network Function Virtualization,” in Proceedings of European Conference and Exhibition on Optical Communications (ECOC 2014), (Optical Society of America, 2014), paper We.3.2.2.

J. Fernandez-Palacios, V. López, B. Cruz, and O. Dios, “Elastic Optical Networking: An Operators Perspective,” in Proceedings of European Conference and Exhibition on Optical Communications (ECOC 2014), (Optical Society of America, 2014), paper Mo.4.2.1.

R. Martínez, R. Casellas, R. Vilalta, and R. Muñoz, “Experimental assessment of GMPLS/PCE-controlled multi-flow optical transponders in flexgrid networks,” in Proceedings of Optical Fiber Communication Conference (OFC 2015), (Optical Society of America, 2015), paper Tu2B.4.
[Crossref]

S. Das, G. Parulkar, and N. McKeown, “Why OpenFlow/SDN can succeed where GMPLS failed,” in Proceedings of European Conference on Optical Communication (ECOC 2012), (Optical Society of America, 2012), paper Tu.1.D.1.
[Crossref]

F. Paolucci, F. Cugini, N. Hussain, F. Fresi, and L. Poti, “OpenFlow-based flexible optical networks with enhanced monitoring functionalities,” in Proceedings of European Conference and Exhibition on Optical Communications (ECOC 2012), (Optical Society of America, 2012), paper Tu.1.D.5.
[Crossref]

C. Wang, W. Hung, and C. Yang, “A prediction based energy conserving resources allocation scheme for cloud computing,” in Proceedings of IEEE International Conference on Granular Computing (GrC 2014), (IEEE, 2014), paper 320–324.
[Crossref]

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Fig. 1 The architecture of MSRI based on NFV for software defined elastic data center optical network.

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Fig. 2 The functional models of network and application resource controllers.

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Fig. 3 The schematic diagram of RIM.

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Fig. 4 Conceptualized illustration of path cascading degree.

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Fig. 5 Flowchart of RIM scheme.

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Fig. 6 Experimental testbed for MSRI and demonstrator setup.

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Fig. 7 (a) Wireshark capture of message sequence for MSRI, (b) extended flow mod message in NC.

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Fig. 8 Comparison on (a) path blocking probability, (b) resource occupation rate and (c) path provisioning latency among various schemes in heavy traffic load scenario.

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Fig. 9 Network topology of NSFNet.

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Tables (1)

Table 1 Notations and Definitions

View Table

Equations (6)

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\begin{array}{l} \begin{matrix}  \end{matrix} S O \begin{matrix} : \begin{matrix}  \end{matrix} \end{matrix} \begin{matrix} S O_{1} & S O_{2} & ... & S O_{i} & ... & S O_{F} \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix} L_{l} \end{matrix} \\ P O_{R} = [\begin{matrix} a_{1, 1} & a_{1, 2} & ... & a_{1, i} & ... & a_{1, F} \\ a_{2, 1} & a_{2, 2} & ... & a_{2, i} & ... & a_{2, F} \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{n_{l}, 1} & a_{n_{l}, 2} & ... & a_{n_{l}, i} & ... & a_{n_{l}, F} \end{matrix}] \begin{matrix}  \end{matrix} \begin{matrix} L_{1} \\ L_{2} \\ ⋮ \\ L_{n_{l}} \end{matrix} \end{array}

\begin{array}{l} R O_{R} = L_{1} \cap L_{2} \cap ... \cap L_{n_{l}} = [\begin{matrix} a_{1}^{R} & a_{2}^{R} & ... & a_{i}^{R} & ... & a_{F}^{R} \end{matrix}] \\ = [\begin{matrix} a_{1, 1} a_{2, 1} ... a_{n_{l}, 1} & a_{1, 2} a_{2, 2} ... a_{n_{l}, 2} & ... & a_{1, i} a_{2, i} ... a_{n_{l}, i} & ... & a_{1, F} a_{2, F} ... a_{n_{l}, F} \end{matrix}] \end{array}

b_{i}^{l} = \sum_{k = 1}^{m_{i}^{l}} b_{k} / m_{i}^{l}

e_{l} = [\begin{matrix} b_{1}^{l} / v_{1}^{l} & b_{2}^{l} / v_{2}^{l} & ... & b_{i}^{l} / v_{i}^{l} & ... & b_{F}^{l} / v_{F}^{l} \end{matrix}]

\begin{array}{l} D_{R} = \frac{cov (e_{R}, {\bar{e}}_{l})}{D (e_{R}) \cdot D ({\bar{e}}_{l})} = \frac{E (e_{R} \cdot {\bar{e}}_{l}) - E (e_{R}) \cdot E ({\bar{e}}_{l})}{\sqrt{E (e_{R}^{2}) - {[E (e_{R})]}^{2}} \cdot \sqrt{E ({\bar{e}}_{l}^{2}) - {[E ({\bar{e}}_{l})]}^{2}}}, \\ E (e_{R}^{2}) - {[E (e_{R})]}^{2} > 0, E ({\bar{e}}_{l}^{2}) - {[E ({\bar{e}}_{l})]}^{2} > 0 \end{array}

\begin{array}{l} e_{R} = [\begin{matrix} b_{1}^{R} / v_{1}^{R} & b_{2}^{R} / v_{2}^{R} & ... & b_{i}^{R} / v_{i}^{R} & ... & b_{F}^{R} / v_{F}^{R} \end{matrix}] \\ \begin{matrix}  \end{matrix} {\bar{e}}_{l} = \sum_{l = 1}^{n_{l}} e_{l} / n_{l} \end{array}

Symbol	Meaning
·R_{s, d}	The route from source s to destination node d
·l_{i, j}	The link between node i and j
·n_l	The number of link on the path R_{s, d}
·DD	The differential delay between multiple fiber-level paths
·PO_R	Link spectrum occupation matrix on R_{s, d} with n_l link
·RO_R	Path spectrum occupation vector on the path R_{s, d}
· $b_{i}^{l}$	The average bandwidth of all probable allocation status using the ith sub-carrier on link l
·e_l	Link spectrum integration eigenvector on link l
·S_p	Link spectrum integration eigenvector space
· ${\bar{e}}_{l}$	Mean vector of link spectrum integration eigenvector space S_p
·GB	The number of sub-carriers are assigned between adjacent spectral flows