Abstract

Creating datacenter federations allows balancing workloads accommodating spikes in cloud demand while reducing the extensive energy consumption of datacenters. This allows datacenter operators to reduce cost and increase their revenue from using under-utilized resources. To implement those elastic operations, datacenters need to be connected through a network providing huge capacity and guaranteeing adequate resource availability when it is required. Those features can be covered by the flexgrid optical technology provided with an intelligent control plane. For the latter, in this paper we rely on the application-based network operations (ABNO) architecture currently under standardization in the IETF. To increase the availability of the desired capacity, a carrier software defined network (SDN) is placed on top of ABNO. On this architecture, cloud resource managers request data transferences using an application-oriented semantic, which includes data volume and required completion time. Those requests are transformed into connection requests by the carrier SDN. To increase resource availability for incoming requests, the carrier SDN can perform elastic operations on already established connections supporting transferences provided that the committed completion time is ensured. Requests are routed and resource allocation is scheduled, which becomes the routing and scheduled spectrum allocation (RSSA) problem. An integer linear programming model is proposed, and an algorithm for solving the RSSA problem in realistic scenarios is designed. Results showing remarkable gains in the amount of data transported motivate its use for inter-datacenter traffic.

© 2014 Optical Society of America

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2014 (2)

L. Velasco, D. King, O. Gerstel, R. Casellas, A. Castro, and V. López, “In-operation network planning,” IEEE Commun. Mag., vol.  52, no. 1, pp. 52–60, 2014.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, A. Castro, and V. López, “Towards a carrier SDN: An example for elastic inter-datacenter connectivity,” Opt. Express, vol.  22, no. 1, pp. 55–61, 2014.
[CrossRef]

2013 (4)

M. Ruiz, M. Pióro, M. Zotkiewicz, M. Klinkowski, and L. Velasco, “Column generation algorithm for RSA problems in flexgrid optical networks,” Photon. Netw. Commun., vol.  26, no. 2–3, pp. 53–64, 2013.
[CrossRef]

W. Lu and Z. Zhu, “Dynamic service provisioning of advance reservation requests in elastic optical networks,” J. Lightwave Technol., vol.  31, no. 10, pp. 1621–1627, 2013.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, V. López, D. Carrera, A. Castro, and J. P. Fernández-Palacios, “Cross-stratum orchestration and flexgrid optical networks for datacenter federations,” IEEE Netw. Mag., vol.  27, no. 6, pp. 23–30, 2013.

M. Klinkowski, M. Ruiz, L. Velasco, D. Careglio, V. Lopez, and J. Comellas, “Elastic spectrum allocation for time-varying traffic in flexgrid optical networks,” IEEE J. Sel. Areas Commun., vol.  31, no. 1, pp. 26–38, 2013.
[CrossRef]

2012 (2)

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag., vol.  50, no. 5, pp. 56–65, 2012.
[CrossRef]

A. Castro, L. Velasco, M. Ruiz, M. Klinkowski, J. P. Fernández-Palacios, and D. Careglio, “Dynamic routing and spectrum (re)allocation in future flexgrid optical networks,” Comput. Netw., vol.  56, no. 12, pp. 2869–2883, 2012.
[CrossRef]

2010 (1)

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

2009 (1)

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: Architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, 2009.
[CrossRef]

Armbrust, M.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

Asensio, A.

L. Velasco, A. Asensio, J. L. Berral, A. Castro, and V. López, “Towards a carrier SDN: An example for elastic inter-datacenter connectivity,” Opt. Express, vol.  22, no. 1, pp. 55–61, 2014.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, V. López, D. Carrera, A. Castro, and J. P. Fernández-Palacios, “Cross-stratum orchestration and flexgrid optical networks for datacenter federations,” IEEE Netw. Mag., vol.  27, no. 6, pp. 23–30, 2013.

A. Asensio, M. Klinkowski, M. Ruiz, V. López, A. Castro, L. Velasco, and J. Comellas, “Impact of aggregation level on the performance of dynamic lightpath adaptation under time-varying traffic,” in Proc. IEEE Optical Network Design and Modeling (ONDM), 2013.

Berral, J. L.

L. Velasco, A. Asensio, J. L. Berral, A. Castro, and V. López, “Towards a carrier SDN: An example for elastic inter-datacenter connectivity,” Opt. Express, vol.  22, no. 1, pp. 55–61, 2014.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, V. López, D. Carrera, A. Castro, and J. P. Fernández-Palacios, “Cross-stratum orchestration and flexgrid optical networks for datacenter federations,” IEEE Netw. Mag., vol.  27, no. 6, pp. 23–30, 2013.

Careglio, D.

M. Klinkowski, M. Ruiz, L. Velasco, D. Careglio, V. Lopez, and J. Comellas, “Elastic spectrum allocation for time-varying traffic in flexgrid optical networks,” IEEE J. Sel. Areas Commun., vol.  31, no. 1, pp. 26–38, 2013.
[CrossRef]

A. Castro, L. Velasco, M. Ruiz, M. Klinkowski, J. P. Fernández-Palacios, and D. Careglio, “Dynamic routing and spectrum (re)allocation in future flexgrid optical networks,” Comput. Netw., vol.  56, no. 12, pp. 2869–2883, 2012.
[CrossRef]

Carrera, D.

L. Velasco, A. Asensio, J. L. Berral, V. López, D. Carrera, A. Castro, and J. P. Fernández-Palacios, “Cross-stratum orchestration and flexgrid optical networks for datacenter federations,” IEEE Netw. Mag., vol.  27, no. 6, pp. 23–30, 2013.

Casellas, R.

L. Velasco, D. King, O. Gerstel, R. Casellas, A. Castro, and V. López, “In-operation network planning,” IEEE Commun. Mag., vol.  52, no. 1, pp. 52–60, 2014.
[CrossRef]

Castro, A.

L. Velasco, D. King, O. Gerstel, R. Casellas, A. Castro, and V. López, “In-operation network planning,” IEEE Commun. Mag., vol.  52, no. 1, pp. 52–60, 2014.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, A. Castro, and V. López, “Towards a carrier SDN: An example for elastic inter-datacenter connectivity,” Opt. Express, vol.  22, no. 1, pp. 55–61, 2014.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, V. López, D. Carrera, A. Castro, and J. P. Fernández-Palacios, “Cross-stratum orchestration and flexgrid optical networks for datacenter federations,” IEEE Netw. Mag., vol.  27, no. 6, pp. 23–30, 2013.

A. Castro, L. Velasco, M. Ruiz, M. Klinkowski, J. P. Fernández-Palacios, and D. Careglio, “Dynamic routing and spectrum (re)allocation in future flexgrid optical networks,” Comput. Netw., vol.  56, no. 12, pp. 2869–2883, 2012.
[CrossRef]

L. Velasco, A. Castro, M. Ruiz, and G. Junyent, “Solving routing and spectrum allocation related optimization problems: From off-line to in-operation flexgrid network planning,” J. Lightwave Technol., to be published.
[CrossRef]

A. Asensio, M. Klinkowski, M. Ruiz, V. López, A. Castro, L. Velasco, and J. Comellas, “Impact of aggregation level on the performance of dynamic lightpath adaptation under time-varying traffic,” in Proc. IEEE Optical Network Design and Modeling (ONDM), 2013.

Comellas, J.

M. Klinkowski, M. Ruiz, L. Velasco, D. Careglio, V. Lopez, and J. Comellas, “Elastic spectrum allocation for time-varying traffic in flexgrid optical networks,” IEEE J. Sel. Areas Commun., vol.  31, no. 1, pp. 26–38, 2013.
[CrossRef]

A. Asensio, M. Klinkowski, M. Ruiz, V. López, A. Castro, L. Velasco, and J. Comellas, “Impact of aggregation level on the performance of dynamic lightpath adaptation under time-varying traffic,” in Proc. IEEE Optical Network Design and Modeling (ONDM), 2013.

Fernández-Palacios, J. P.

L. Velasco, A. Asensio, J. L. Berral, V. López, D. Carrera, A. Castro, and J. P. Fernández-Palacios, “Cross-stratum orchestration and flexgrid optical networks for datacenter federations,” IEEE Netw. Mag., vol.  27, no. 6, pp. 23–30, 2013.

A. Castro, L. Velasco, M. Ruiz, M. Klinkowski, J. P. Fernández-Palacios, and D. Careglio, “Dynamic routing and spectrum (re)allocation in future flexgrid optical networks,” Comput. Netw., vol.  56, no. 12, pp. 2869–2883, 2012.
[CrossRef]

Fox, A.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

Gerstel, O.

L. Velasco, D. King, O. Gerstel, R. Casellas, A. Castro, and V. López, “In-operation network planning,” IEEE Commun. Mag., vol.  52, no. 1, pp. 52–60, 2014.
[CrossRef]

Goiri, I.

I. Goiri, J. Guitart, and J. Torres, “Characterizing cloud federation for enhancing providers’ profit,” in Proc. IEEE Int. Conf. on Cloud Computing, 2010.

Griffith, R.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

Guitart, J.

I. Goiri, J. Guitart, and J. Torres, “Characterizing cloud federation for enhancing providers’ profit,” in Proc. IEEE Int. Conf. on Cloud Computing, 2010.

Hirano, A.

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag., vol.  50, no. 5, pp. 56–65, 2012.
[CrossRef]

Jinno, M.

M. Jinno, H. Takara, Y. Sone, K. Yonenaga, and A. Hirano, “Multiflow optical transponder for efficient multilayer optical networking,” IEEE Commun. Mag., vol.  50, no. 5, pp. 56–65, 2012.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: Architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, 2009.
[CrossRef]

Joseph, A. D.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

Junyent, G.

L. Velasco, A. Castro, M. Ruiz, and G. Junyent, “Solving routing and spectrum allocation related optimization problems: From off-line to in-operation flexgrid network planning,” J. Lightwave Technol., to be published.
[CrossRef]

Katz, R. H.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

King, D.

L. Velasco, D. King, O. Gerstel, R. Casellas, A. Castro, and V. López, “In-operation network planning,” IEEE Commun. Mag., vol.  52, no. 1, pp. 52–60, 2014.
[CrossRef]

Klinkowski, M.

M. Ruiz, M. Pióro, M. Zotkiewicz, M. Klinkowski, and L. Velasco, “Column generation algorithm for RSA problems in flexgrid optical networks,” Photon. Netw. Commun., vol.  26, no. 2–3, pp. 53–64, 2013.
[CrossRef]

M. Klinkowski, M. Ruiz, L. Velasco, D. Careglio, V. Lopez, and J. Comellas, “Elastic spectrum allocation for time-varying traffic in flexgrid optical networks,” IEEE J. Sel. Areas Commun., vol.  31, no. 1, pp. 26–38, 2013.
[CrossRef]

A. Castro, L. Velasco, M. Ruiz, M. Klinkowski, J. P. Fernández-Palacios, and D. Careglio, “Dynamic routing and spectrum (re)allocation in future flexgrid optical networks,” Comput. Netw., vol.  56, no. 12, pp. 2869–2883, 2012.
[CrossRef]

A. Asensio, M. Klinkowski, M. Ruiz, V. López, A. Castro, L. Velasco, and J. Comellas, “Impact of aggregation level on the performance of dynamic lightpath adaptation under time-varying traffic,” in Proc. IEEE Optical Network Design and Modeling (ONDM), 2013.

Konwinski, A.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

Kozicki, B.

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: Architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, 2009.
[CrossRef]

Lee, G.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

Lopez, V.

M. Klinkowski, M. Ruiz, L. Velasco, D. Careglio, V. Lopez, and J. Comellas, “Elastic spectrum allocation for time-varying traffic in flexgrid optical networks,” IEEE J. Sel. Areas Commun., vol.  31, no. 1, pp. 26–38, 2013.
[CrossRef]

López, V.

L. Velasco, D. King, O. Gerstel, R. Casellas, A. Castro, and V. López, “In-operation network planning,” IEEE Commun. Mag., vol.  52, no. 1, pp. 52–60, 2014.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, A. Castro, and V. López, “Towards a carrier SDN: An example for elastic inter-datacenter connectivity,” Opt. Express, vol.  22, no. 1, pp. 55–61, 2014.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, V. López, D. Carrera, A. Castro, and J. P. Fernández-Palacios, “Cross-stratum orchestration and flexgrid optical networks for datacenter federations,” IEEE Netw. Mag., vol.  27, no. 6, pp. 23–30, 2013.

A. Asensio, M. Klinkowski, M. Ruiz, V. López, A. Castro, L. Velasco, and J. Comellas, “Impact of aggregation level on the performance of dynamic lightpath adaptation under time-varying traffic,” in Proc. IEEE Optical Network Design and Modeling (ONDM), 2013.

Lu, W.

Matsuoka, S.

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: Architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, 2009.
[CrossRef]

Patterson, D. A.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

Pióro, M.

M. Ruiz, M. Pióro, M. Zotkiewicz, M. Klinkowski, and L. Velasco, “Column generation algorithm for RSA problems in flexgrid optical networks,” Photon. Netw. Commun., vol.  26, no. 2–3, pp. 53–64, 2013.
[CrossRef]

Rabkin, A.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[CrossRef]

Ruiz, M.

M. Klinkowski, M. Ruiz, L. Velasco, D. Careglio, V. Lopez, and J. Comellas, “Elastic spectrum allocation for time-varying traffic in flexgrid optical networks,” IEEE J. Sel. Areas Commun., vol.  31, no. 1, pp. 26–38, 2013.
[CrossRef]

M. Ruiz, M. Pióro, M. Zotkiewicz, M. Klinkowski, and L. Velasco, “Column generation algorithm for RSA problems in flexgrid optical networks,” Photon. Netw. Commun., vol.  26, no. 2–3, pp. 53–64, 2013.
[CrossRef]

A. Castro, L. Velasco, M. Ruiz, M. Klinkowski, J. P. Fernández-Palacios, and D. Careglio, “Dynamic routing and spectrum (re)allocation in future flexgrid optical networks,” Comput. Netw., vol.  56, no. 12, pp. 2869–2883, 2012.
[CrossRef]

A. Asensio, M. Klinkowski, M. Ruiz, V. López, A. Castro, L. Velasco, and J. Comellas, “Impact of aggregation level on the performance of dynamic lightpath adaptation under time-varying traffic,” in Proc. IEEE Optical Network Design and Modeling (ONDM), 2013.

L. Velasco, A. Castro, M. Ruiz, and G. Junyent, “Solving routing and spectrum allocation related optimization problems: From off-line to in-operation flexgrid network planning,” J. Lightwave Technol., to be published.
[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., vol.  50, no. 5, pp. 56–65, 2012.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: Architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, 2009.
[CrossRef]

Stoica, I.

M. Armbrust, A. Fox, R. Griffith, A. D. Joseph, R. H. Katz, A. Konwinski, G. Lee, D. A. Patterson, A. Rabkin, I. Stoica, and M. Zaharia, “A view of cloud computing,” Commun. ACM, vol.  53, no. 4, pp. 50–58, 2010.
[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., vol.  50, no. 5, pp. 56–65, 2012.
[CrossRef]

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: Architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, 2009.
[CrossRef]

Torres, J.

I. Goiri, J. Guitart, and J. Torres, “Characterizing cloud federation for enhancing providers’ profit,” in Proc. IEEE Int. Conf. on Cloud Computing, 2010.

Tsukishima, Y.

M. Jinno, H. Takara, B. Kozicki, Y. Tsukishima, Y. Sone, and S. Matsuoka, “Spectrum-efficient and scalable elastic optical path network: Architecture, benefits, and enabling technologies,” IEEE Commun. Mag., vol.  47, no. 11, pp. 66–73, 2009.
[CrossRef]

Velasco, L.

L. Velasco, D. King, O. Gerstel, R. Casellas, A. Castro, and V. López, “In-operation network planning,” IEEE Commun. Mag., vol.  52, no. 1, pp. 52–60, 2014.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, A. Castro, and V. López, “Towards a carrier SDN: An example for elastic inter-datacenter connectivity,” Opt. Express, vol.  22, no. 1, pp. 55–61, 2014.
[CrossRef]

M. Ruiz, M. Pióro, M. Zotkiewicz, M. Klinkowski, and L. Velasco, “Column generation algorithm for RSA problems in flexgrid optical networks,” Photon. Netw. Commun., vol.  26, no. 2–3, pp. 53–64, 2013.
[CrossRef]

M. Klinkowski, M. Ruiz, L. Velasco, D. Careglio, V. Lopez, and J. Comellas, “Elastic spectrum allocation for time-varying traffic in flexgrid optical networks,” IEEE J. Sel. Areas Commun., vol.  31, no. 1, pp. 26–38, 2013.
[CrossRef]

L. Velasco, A. Asensio, J. L. Berral, V. López, D. Carrera, A. Castro, and J. P. Fernández-Palacios, “Cross-stratum orchestration and flexgrid optical networks for datacenter federations,” IEEE Netw. Mag., vol.  27, no. 6, pp. 23–30, 2013.

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Figures (7)

Fig. 1.
Fig. 1.

Control architecture scheme. (a) Carrier SDN translates application-oriented and connection-oriented semantics. (b) Example of messages exchanged to request connection operations.

Fig. 2.
Fig. 2.

Scheduling schemes for spectrum allocation. (a) Bitrate squeezing only and (b) with scheduled resource reservation.

Fig. 3.
Fig. 3.

Three feasible reschedulings for a given transference.

Fig. 4.
Fig. 4.

Telefonica (TEL), British Telecom (BT), and Deutsche Telekom (DT) network topologies used in this paper.

Fig. 5.
Fig. 5.

Unserved bitrate and transference completion time versus load (in terms of number of VMs). Only transfer-mode requests are considered.

Fig. 6.
Fig. 6.

Unserved bitrate and transference completion time versus VMs’ size. Only transfer-mode requests are considered.

Fig. 7.
Fig. 7.

Unserved bitrate of transfer-mode requests and transference completion time versus offered load of fixed-bitrate traffic.

Tables (6)

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TABLE I Algorithm for Computing Rectangle Sets

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TABLE II Algorithm for the RSSA Problem

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TABLE III Algorithm for Resource Assignment

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TABLE IV ILP Model Versus Heuristic Comparison

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TABLE V Offered Load Gain and Transference Time Comparison

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TABLE VI Elastic Operations at 1% of Unserved Bitrate

Equations (7)

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( RSSA ) minimize r R y r
a A p P x a p = 1 ,
a A 0 ( r ) x a r 0 = 1 r R ,
a A 1 ( r ) x a r 1 = 1 r R ,
a A 1 ( r ) β r a a x a r 1 x a r 0 r R , a A 0 ( r ) ,
x a r 0 ω a r y r r R , a A 0 ( r ) ,
r R a A 0 ( r ) δ a s · γ a t · ρ r e · x a r 0 + r R a A 1 ( r ) δ a s · γ a t · ρ r e · x a r 1 + a A p P δ a s · γ a t · ρ p e · x a p α e s t e E , s S , t { t 0 , , T } .