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.

© 2015 Optical Society of America

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References

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  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]
  2. C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Comm. Surv. and Tutor. 14(4), 1021–1036 (2012).
    [Crossref]
  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]
  4. 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]
  5. W. Shieh, “OFDM for flexible high-speed optical networks,” J. Lightwave Technol. 29(10), 1560–1577 (2011).
    [Crossref]
  6. 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]
  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.
  8. 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]
  9. 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.
  10. 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]
  11. 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.
  12. 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]
  13. 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]
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. 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]
  21. 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]
  22. 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)

2014 (4)

2013 (2)

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)

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.

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]

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.

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]

Beloglazov, A.

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]

Buyya, R.

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]

Careglio, D.

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]

Casellas, R.

Cervello-Pastor, C.

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]

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.

Hirano, A.

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]

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.

Jinno, M.

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]

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.

Leivadeas, A.

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]

Li, G.

Li, H.

Lin, Y.

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.

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]

Martínez, R.

Monje, A.

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]

Morita, I.

Muñoz, R.

Nejabati, R.

Palkopoulou, E.

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]

Papagianni, C.

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]

Papavassiliou, S.

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]

Peng, S.

Peng, W. R.

Rashidi Fard, M.

Shieh, W.

Simeonidou, D.

Sole-Pareta, J.

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]

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.

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]

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.

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.

Zervas, G.

Zhang, J.

Zhao, Y.

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)

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]

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)

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]

J. Lightwave Technol. (1)

Opt. Express (6)

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]

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]

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]

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]

Proc. IEEE (1)

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]

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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Figures (9)

Fig. 1
Fig. 1 The architecture of MSRI based on NFV for software defined elastic data center optical network.
Fig. 2
Fig. 2 The functional models of network and application resource controllers.
Fig. 3
Fig. 3 The schematic diagram of RIM.
Fig. 4
Fig. 4 Conceptualized illustration of path cascading degree.
Fig. 5
Fig. 5 Flowchart of RIM scheme.
Fig. 6
Fig. 6 Experimental testbed for MSRI and demonstrator setup.
Fig. 7
Fig. 7 (a) Wireshark capture of message sequence for MSRI, (b) extended flow mod message in NC.
Fig. 8
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.
Fig. 9
Fig. 9 Network topology of NSFNet.

Tables (1)

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Table 1 Notations and Definitions

Equations (6)

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

SO : S O 1 S O 2 ... S O i ... S O F L l P O R =[ 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 ] L 1 L 2 L n l
R O R = L 1 L 2 ... L n l =[ a 1 R a 2 R ... a i R ... a F R ] =[ 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 ]
b i l = k=1 m i l b k / m i l
e l =[ b 1 l / v 1 l b 2 l / v 2 l ... b i l / v i l ... b F l / v F l ]
D R = cov( e R , e ¯ l ) D( e R )D( e ¯ l ) = E( e R e ¯ l )E( e R )E( e ¯ l ) E( e R 2 ) [ E( e R ) ] 2 E( e ¯ l 2 ) [ E( e ¯ l ) ] 2 , E( e R 2 ) [ E( e R ) ] 2 >0,E( e ¯ l 2 ) [ E( e ¯ l ) ] 2 >0
e R =[ b 1 R / v 1 R b 2 R / v 2 R ... b i R / v i R ... b F R / v F R ] e ¯ l = l=1 n l e l / n l

Metrics