Abstract

Cloud services are fundamentally supported by data center networks (DCNs). With the fast growth of cloud services, the scale of DCNs is increasing rapidly, leading to great concern about system scalability due to multiple constraints. This paper proposes a scalable DCN architecture based on optical switching and transmission, with the distributed placement of optical switches and server racks at different nodes in a given optical network. This solves the scalability issue by relaxing power and cooling constraints and by reducing the number of (electronic) switches using high-capacity optical switches, as well as by simplifying DCN internal connections using wavelengths in the optical network. Moreover, the distributed optical switches provide service access interfaces to meet demand within areas, and thus reduce the transmission cost of the external traffic. The major concern is the additional delay and cost for remote transmissions of the DCN internal traffic. To this end, we study the component placement problem in DCNs under a given set of external demands and internal traffic patterns. By leveraging among multiple conflicting factors such as scalability and internal overhead of the DCN as well as the transmission cost of external traffic, we propose both an integer linear program and a heuristic to minimize the system cost of a DCN while satisfying all service demands in the network. This addresses both scalability and cost minimization issues from a network point of view.

© 2014 Optical Society of America

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2013 (3)

Y. Zhang and N. Ansari, “On architecture design, congestion notification, TCP incast and power consumption in data centers,” IEEE Commun. Surv. Tutorials, vol.  15, no. 1, pp. 39–64, 2013.
[CrossRef]

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

L. Liu, D. Zhang, T. Tsuritani, R. Vilalta, R. Casellas, L. Hong, I. Morita, H. Guo, J. Wu, R. Martinez, and R. Munoz, “Field trial of an OpenFlow-based unified control plane for multi-layer multi-granularity optical switching networks,” J. Lightwave Technol., vol.  31, no. 4, pp. 506–514, 2013.
[CrossRef]

2012 (3)

Z. Zheng, T. Zhou, M. Lyu, and I. King, “Component ranking for fault-tolerant cloud applications,” IEEE Trans. Serv. Comput., vol.  5, no. 4, pp. 540–550, 2012.

C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Commun. Surv. Tutorials, vol.  14, no. 4, pp. 1021–1036, 2012.
[CrossRef]

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

2011 (1)

F. O. Sem-Jacobsen, T. Skeie, O. Lysne, and J. Duato, “Dynamic fault tolerance in fat trees,” IEEE Trans. Comput., vol.  60, no. 4, pp. 508–525, 2011.
[CrossRef]

2010 (1)

C. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag., vol.  48, no. 7, pp. 32–39, July 2010.
[CrossRef]

2009 (6)

L. Vaquero, L. Rodero-Merino, J. Caceres, and M. Lindner, “A break in the clouds: Towards a cloud definition,” Comput. Commun. Rev., vol.  39, no. 1, pp. 50–55, Jan. 2009.
[CrossRef]

B. Wu, K. L. Yeung, P.-H. Ho, and X. H. Jiang, “Minimum delay scheduling for performance guaranteed switches with optical fabrics,” J. Lightwave Technol., vol.  27, no. 16, pp. 3453–3465, Aug. 2009.
[CrossRef]

B. Wu, K. L. Yeung, M. Hamdi, and X. Li, “Minimizing internal speedup for performance guaranteed switches with optical fabrics,” IEEE/ACM Trans. Netw., vol.  17, no. 2, pp. 632–645, Apr. 2009.
[CrossRef]

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

X. Yuan, W. Nienaber, Z. Duan, and R. Melhem, “Oblivious routing in fat-tree based system area networks with uncertain traffic demands,” IEEE/ACM Trans. Netw., vol.  17, no. 5, pp. 1439–1452, 2009.
[CrossRef]

S. Coll, F. J. Mora, J. Duato, and F. Petrini, “Efficient and scalable hardware-based multicast in fat-tree networks,” IEEE Trans. Parallel Distrib. Syst., vol.  20, no. 9, pp. 1285–1298, 2009.

2008 (2)

C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, “DCell: A scalable and fault-tolerant network structure for data centers,” Comput. Commun. Rev., vol.  38, no. 4, pp. 75–86, Oct. 2008.
[CrossRef]

F. Wang and M. Hamdi, “Strictly non-blocking conditions for the central-stage buffered Clos-network,” IEEE Commun. Lett., vol.  12, no. 3, pp. 206–208, 2008.
[CrossRef]

2005 (1)

2003 (2)

H. J. Chao, Z. Jing, and S. Y. Liew, “Matching algorithms for three-stage bufferless Clos network switches,” IEEE Commun. Mag., vol.  41, no. 10, pp. 46–54, 2003.
[CrossRef]

B. Towles and W. J. Dally, “Guaranteed scheduling for switches with configuration overhead,” IEEE/ACM Trans. Netw., vol.  11, no. 5, pp. 835–847, Oct. 2003.
[CrossRef]

Akella, V.

X. Ye, Y. Yin, S. J. B. Yoo, P. Mejia, R. Proietti, and V. Akella, “DOS: A scalable optical switch for datacenters,” in Proc. 6th ACM/IEEE Symp. on Architectures for Networking and Communications Systems (ANCS), 2010, p. 24.

Andersen, D. G.

G. Wang, D. G. Andersen, M. Kaminsky, K. Papagiannaki, T. E. Ng, M. Kozuch, and M. Ryan, “c-Through: Part-time optics in data centers,” in Proc. ACM SIGCOMM, 2010, pp. 327–338.

Ansari, N.

Y. Zhang and N. Ansari, “On architecture design, congestion notification, TCP incast and power consumption in data centers,” IEEE Commun. Surv. Tutorials, vol.  15, no. 1, pp. 39–64, 2013.
[CrossRef]

Y. Zhang and N. Ansari, “HERO: Hierarchical energy optimization for data center networks,” IEEE Syst. J., to be published.

Y. Zhang and N. Ansari, “On mitigating TCP incast in data center networks,” IEEE INFOCOM, Shanghai, 2011, pp. 51–55.

Bari, M.

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

Bazzaz, H. H.

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

Benner, A.

A. Benner, “Optical interconnect opportunities in supercomputers and high end computing,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), 2012, paper OTu2B.4.

Bernasconi, P.

J. Gripp, J. E. Simsarian, J. D. LeGrange, P. Bernasconi, and D. T. Neilson, “Photonic terabit routers: The IRIS project,” in Optical Fiber Communication Conf., 2010, paper OThP3.

Boutaba, R.

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

Caceres, J.

L. Vaquero, L. Rodero-Merino, J. Caceres, and M. Lindner, “A break in the clouds: Towards a cloud definition,” Comput. Commun. Rev., vol.  39, no. 1, pp. 50–55, Jan. 2009.
[CrossRef]

Casellas, R.

Chao, H. J.

S. Jiang, G. Hu, S. Y. Liew, and H. J. Chao, “Scheduling algorithms for shared fiber-delay-line optical packet switches—Part II: The three-stage Clos-network case,” J. Lightwave Technol., vol.  23, no. 4, pp. 1601–1609, 2005.
[CrossRef]

H. J. Chao, Z. Jing, and S. Y. Liew, “Matching algorithms for three-stage bufferless Clos network switches,” IEEE Commun. Mag., vol.  41, no. 10, pp. 46–54, 2003.
[CrossRef]

K. Xi, Y.-H. Kao, M. Yang, and H. J. Chao, “Petabit optical switch for data center networks,” Tech. Rep., Polytechnic Institute of New York University, 2010.

Chen, K.

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

Chen, Y.

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

Coll, S.

S. Coll, F. J. Mora, J. Duato, and F. Petrini, “Efficient and scalable hardware-based multicast in fat-tree networks,” IEEE Trans. Parallel Distrib. Syst., vol.  20, no. 9, pp. 1285–1298, 2009.

Dally, W. J.

B. Towles and W. J. Dally, “Guaranteed scheduling for switches with configuration overhead,” IEEE/ACM Trans. Netw., vol.  11, no. 5, pp. 835–847, Oct. 2003.
[CrossRef]

Duan, Z.

X. Yuan, W. Nienaber, Z. Duan, and R. Melhem, “Oblivious routing in fat-tree based system area networks with uncertain traffic demands,” IEEE/ACM Trans. Netw., vol.  17, no. 5, pp. 1439–1452, 2009.
[CrossRef]

Duato, J.

F. O. Sem-Jacobsen, T. Skeie, O. Lysne, and J. Duato, “Dynamic fault tolerance in fat trees,” IEEE Trans. Comput., vol.  60, no. 4, pp. 508–525, 2011.
[CrossRef]

S. Coll, F. J. Mora, J. Duato, and F. Petrini, “Efficient and scalable hardware-based multicast in fat-tree networks,” IEEE Trans. Parallel Distrib. Syst., vol.  20, no. 9, pp. 1285–1298, 2009.

Esteves, R.

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

Fainman, Y.

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

Farrington, N.

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

Feng, Z.

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

Gill, V.

C. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag., vol.  48, no. 7, pp. 32–39, July 2010.
[CrossRef]

Granville, L.

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

Gripp, J.

J. Gripp, J. E. Simsarian, J. D. LeGrange, P. Bernasconi, and D. T. Neilson, “Photonic terabit routers: The IRIS project,” in Optical Fiber Communication Conf., 2010, paper OThP3.

Guo, C.

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, “DCell: A scalable and fault-tolerant network structure for data centers,” Comput. Commun. Rev., vol.  38, no. 4, pp. 75–86, Oct. 2008.
[CrossRef]

Guo, H.

Hamdi, M.

B. Wu, K. L. Yeung, M. Hamdi, and X. Li, “Minimizing internal speedup for performance guaranteed switches with optical fabrics,” IEEE/ACM Trans. Netw., vol.  17, no. 2, pp. 632–645, Apr. 2009.
[CrossRef]

F. Wang and M. Hamdi, “Strictly non-blocking conditions for the central-stage buffered Clos-network,” IEEE Commun. Lett., vol.  12, no. 3, pp. 206–208, 2008.
[CrossRef]

Harmer, T.

P. Wright, T. Harmer, J. Hawkins, and Y. L. Sun, “A commodity-focused multi-cloud marketplace exemplar application,” in 2011 IEEE Int. Conf. on Cloud Computing (CLOUD), 2011, pp. 590–597.

Hawkins, J.

P. Wright, T. Harmer, J. Hawkins, and Y. L. Sun, “A commodity-focused multi-cloud marketplace exemplar application,” in 2011 IEEE Int. Conf. on Cloud Computing (CLOUD), 2011, pp. 590–597.

Ho, P.-H.

Hong, L.

Hu, G.

Huang, N.

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

Jiang, S.

Jiang, X. H.

Jing, Z.

H. J. Chao, Z. Jing, and S. Y. Liew, “Matching algorithms for three-stage bufferless Clos network switches,” IEEE Commun. Mag., vol.  41, no. 10, pp. 46–54, 2003.
[CrossRef]

Kachris, C.

C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Commun. Surv. Tutorials, vol.  14, no. 4, pp. 1021–1036, 2012.
[CrossRef]

Kamalov, V.

C. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag., vol.  48, no. 7, pp. 32–39, July 2010.
[CrossRef]

Kaminsky, M.

G. Wang, D. G. Andersen, M. Kaminsky, K. Papagiannaki, T. E. Ng, M. Kozuch, and M. Ryan, “c-Through: Part-time optics in data centers,” in Proc. ACM SIGCOMM, 2010, pp. 327–338.

Kandlur, D. D.

D. D. Kandlur and T. W. Keller, “Green data centers and hot chips,” in Proc. 46th Annu. Design Automation Conf., 2009, pp. 888–890.

Kao, Y.-H.

K. Xi, Y.-H. Kao, M. Yang, and H. J. Chao, “Petabit optical switch for data center networks,” Tech. Rep., Polytechnic Institute of New York University, 2010.

Keller, T. W.

D. D. Kandlur and T. W. Keller, “Green data centers and hot chips,” in Proc. 46th Annu. Design Automation Conf., 2009, pp. 888–890.

King, I.

Z. Zheng, T. Zhou, M. Lyu, and I. King, “Component ranking for fault-tolerant cloud applications,” IEEE Trans. Serv. Comput., vol.  5, no. 4, pp. 540–550, 2012.

Koley, B.

C. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag., vol.  48, no. 7, pp. 32–39, July 2010.
[CrossRef]

Kozuch, M.

G. Wang, D. G. Andersen, M. Kaminsky, K. Papagiannaki, T. E. Ng, M. Kozuch, and M. Ryan, “c-Through: Part-time optics in data centers,” in Proc. ACM SIGCOMM, 2010, pp. 327–338.

Lam, C.

C. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag., vol.  48, no. 7, pp. 32–39, July 2010.
[CrossRef]

LeGrange, J. D.

J. Gripp, J. E. Simsarian, J. D. LeGrange, P. Bernasconi, and D. T. Neilson, “Photonic terabit routers: The IRIS project,” in Optical Fiber Communication Conf., 2010, paper OThP3.

Li, X.

B. Wu, K. L. Yeung, M. Hamdi, and X. Li, “Minimizing internal speedup for performance guaranteed switches with optical fabrics,” IEEE/ACM Trans. Netw., vol.  17, no. 2, pp. 632–645, Apr. 2009.
[CrossRef]

Liew, S. Y.

S. Jiang, G. Hu, S. Y. Liew, and H. J. Chao, “Scheduling algorithms for shared fiber-delay-line optical packet switches—Part II: The three-stage Clos-network case,” J. Lightwave Technol., vol.  23, no. 4, pp. 1601–1609, 2005.
[CrossRef]

H. J. Chao, Z. Jing, and S. Y. Liew, “Matching algorithms for three-stage bufferless Clos network switches,” IEEE Commun. Mag., vol.  41, no. 10, pp. 46–54, 2003.
[CrossRef]

Lindner, M.

L. Vaquero, L. Rodero-Merino, J. Caceres, and M. Lindner, “A break in the clouds: Towards a cloud definition,” Comput. Commun. Rev., vol.  39, no. 1, pp. 50–55, Jan. 2009.
[CrossRef]

Liu, H.

C. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag., vol.  48, no. 7, pp. 32–39, July 2010.
[CrossRef]

Liu, L.

Lu, S.

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, “DCell: A scalable and fault-tolerant network structure for data centers,” Comput. Commun. Rev., vol.  38, no. 4, pp. 75–86, Oct. 2008.
[CrossRef]

Lysne, O.

F. O. Sem-Jacobsen, T. Skeie, O. Lysne, and J. Duato, “Dynamic fault tolerance in fat trees,” IEEE Trans. Comput., vol.  60, no. 4, pp. 508–525, 2011.
[CrossRef]

Lyu, M.

Z. Zheng, T. Zhou, M. Lyu, and I. King, “Component ranking for fault-tolerant cloud applications,” IEEE Trans. Serv. Comput., vol.  5, no. 4, pp. 540–550, 2012.

Martinez, R.

Mejia, P.

X. Ye, Y. Yin, S. J. B. Yoo, P. Mejia, R. Proietti, and V. Akella, “DOS: A scalable optical switch for datacenters,” in Proc. 6th ACM/IEEE Symp. on Architectures for Networking and Communications Systems (ANCS), 2010, p. 24.

Melhem, R.

X. Yuan, W. Nienaber, Z. Duan, and R. Melhem, “Oblivious routing in fat-tree based system area networks with uncertain traffic demands,” IEEE/ACM Trans. Netw., vol.  17, no. 5, pp. 1439–1452, 2009.
[CrossRef]

Miri, P.

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

Mora, F. J.

S. Coll, F. J. Mora, J. Duato, and F. Petrini, “Efficient and scalable hardware-based multicast in fat-tree networks,” IEEE Trans. Parallel Distrib. Syst., vol.  20, no. 9, pp. 1285–1298, 2009.

Morita, I.

Munoz, R.

Mysore, R. N.

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

Neilson, D. T.

J. Gripp, J. E. Simsarian, J. D. LeGrange, P. Bernasconi, and D. T. Neilson, “Photonic terabit routers: The IRIS project,” in Optical Fiber Communication Conf., 2010, paper OThP3.

Ng, T. E.

G. Wang, D. G. Andersen, M. Kaminsky, K. Papagiannaki, T. E. Ng, M. Kozuch, and M. Ryan, “c-Through: Part-time optics in data centers,” in Proc. ACM SIGCOMM, 2010, pp. 327–338.

Nienaber, W.

X. Yuan, W. Nienaber, Z. Duan, and R. Melhem, “Oblivious routing in fat-tree based system area networks with uncertain traffic demands,” IEEE/ACM Trans. Netw., vol.  17, no. 5, pp. 1439–1452, 2009.
[CrossRef]

Pamboris, A.

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

Papagiannaki, K.

G. Wang, D. G. Andersen, M. Kaminsky, K. Papagiannaki, T. E. Ng, M. Kozuch, and M. Ryan, “c-Through: Part-time optics in data centers,” in Proc. ACM SIGCOMM, 2010, pp. 327–338.

Papen, G.

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

Petrini, F.

S. Coll, F. J. Mora, J. Duato, and F. Petrini, “Efficient and scalable hardware-based multicast in fat-tree networks,” IEEE Trans. Parallel Distrib. Syst., vol.  20, no. 9, pp. 1285–1298, 2009.

Podlesny, M.

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

Porter, G.

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

Proietti, R.

X. Ye, Y. Yin, S. J. B. Yoo, P. Mejia, R. Proietti, and V. Akella, “DOS: A scalable optical switch for datacenters,” in Proc. 6th ACM/IEEE Symp. on Architectures for Networking and Communications Systems (ANCS), 2010, p. 24.

Rabbani, M.

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

Radhakrishnan, S.

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

Ramachandran, K.

A. Singla, A. Singh, K. Ramachandran, L. Xu, and Y. Zhang, “Proteus: A topology malleable data center network,” in Proc. Ninth ACM SIGCOMM Workshop on Hot Topics in Networks (Hotnets), 2010, p. 8.

Rodero-Merino, L.

L. Vaquero, L. Rodero-Merino, J. Caceres, and M. Lindner, “A break in the clouds: Towards a cloud definition,” Comput. Commun. Rev., vol.  39, no. 1, pp. 50–55, Jan. 2009.
[CrossRef]

Ryan, M.

G. Wang, D. G. Andersen, M. Kaminsky, K. Papagiannaki, T. E. Ng, M. Kozuch, and M. Ryan, “c-Through: Part-time optics in data centers,” in Proc. ACM SIGCOMM, 2010, pp. 327–338.

Sem-Jacobsen, F. O.

F. O. Sem-Jacobsen, T. Skeie, O. Lysne, and J. Duato, “Dynamic fault tolerance in fat trees,” IEEE Trans. Comput., vol.  60, no. 4, pp. 508–525, 2011.
[CrossRef]

Shi, L.

C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, “DCell: A scalable and fault-tolerant network structure for data centers,” Comput. Commun. Rev., vol.  38, no. 4, pp. 75–86, Oct. 2008.
[CrossRef]

Simsarian, J. E.

J. Gripp, J. E. Simsarian, J. D. LeGrange, P. Bernasconi, and D. T. Neilson, “Photonic terabit routers: The IRIS project,” in Optical Fiber Communication Conf., 2010, paper OThP3.

Singh, A.

A. Singla, A. Singh, K. Ramachandran, L. Xu, and Y. Zhang, “Proteus: A topology malleable data center network,” in Proc. Ninth ACM SIGCOMM Workshop on Hot Topics in Networks (Hotnets), 2010, p. 8.

Singla, A.

A. Singla, A. Singh, K. Ramachandran, L. Xu, and Y. Zhang, “Proteus: A topology malleable data center network,” in Proc. Ninth ACM SIGCOMM Workshop on Hot Topics in Networks (Hotnets), 2010, p. 8.

Skeie, T.

F. O. Sem-Jacobsen, T. Skeie, O. Lysne, and J. Duato, “Dynamic fault tolerance in fat trees,” IEEE Trans. Comput., vol.  60, no. 4, pp. 508–525, 2011.
[CrossRef]

Subramanya, V.

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

Sun, Y. L.

P. Wright, T. Harmer, J. Hawkins, and Y. L. Sun, “A commodity-focused multi-cloud marketplace exemplar application,” in 2011 IEEE Int. Conf. on Cloud Computing (CLOUD), 2011, pp. 590–597.

Tan, K.

C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, “DCell: A scalable and fault-tolerant network structure for data centers,” Comput. Commun. Rev., vol.  38, no. 4, pp. 75–86, Oct. 2008.
[CrossRef]

Tomkos, I.

C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Commun. Surv. Tutorials, vol.  14, no. 4, pp. 1021–1036, 2012.
[CrossRef]

Towles, B.

B. Towles and W. J. Dally, “Guaranteed scheduling for switches with configuration overhead,” IEEE/ACM Trans. Netw., vol.  11, no. 5, pp. 835–847, Oct. 2003.
[CrossRef]

Tsuritani, T.

Vahdat, A.

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

Vaquero, L.

L. Vaquero, L. Rodero-Merino, J. Caceres, and M. Lindner, “A break in the clouds: Towards a cloud definition,” Comput. Commun. Rev., vol.  39, no. 1, pp. 50–55, Jan. 2009.
[CrossRef]

Vilalta, R.

Wang, F.

F. Wang and M. Hamdi, “Strictly non-blocking conditions for the central-stage buffered Clos-network,” IEEE Commun. Lett., vol.  12, no. 3, pp. 206–208, 2008.
[CrossRef]

Wang, G.

G. Wang, D. G. Andersen, M. Kaminsky, K. Papagiannaki, T. E. Ng, M. Kozuch, and M. Ryan, “c-Through: Part-time optics in data centers,” in Proc. ACM SIGCOMM, 2010, pp. 327–338.

Wright, P.

P. Wright, T. Harmer, J. Hawkins, and Y. L. Sun, “A commodity-focused multi-cloud marketplace exemplar application,” in 2011 IEEE Int. Conf. on Cloud Computing (CLOUD), 2011, pp. 590–597.

Wu, B.

B. Wu, K. L. Yeung, P.-H. Ho, and X. H. Jiang, “Minimum delay scheduling for performance guaranteed switches with optical fabrics,” J. Lightwave Technol., vol.  27, no. 16, pp. 3453–3465, Aug. 2009.
[CrossRef]

B. Wu, K. L. Yeung, M. Hamdi, and X. Li, “Minimizing internal speedup for performance guaranteed switches with optical fabrics,” IEEE/ACM Trans. Netw., vol.  17, no. 2, pp. 632–645, Apr. 2009.
[CrossRef]

Wu, H.

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, “DCell: A scalable and fault-tolerant network structure for data centers,” Comput. Commun. Rev., vol.  38, no. 4, pp. 75–86, Oct. 2008.
[CrossRef]

Wu, J.

Wu, W.

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

Xi, K.

K. Xi, Y.-H. Kao, M. Yang, and H. J. Chao, “Petabit optical switch for data center networks,” Tech. Rep., Polytechnic Institute of New York University, 2010.

Xu, L.

A. Singla, A. Singh, K. Ramachandran, L. Xu, and Y. Zhang, “Proteus: A topology malleable data center network,” in Proc. Ninth ACM SIGCOMM Workshop on Hot Topics in Networks (Hotnets), 2010, p. 8.

Yang, M.

K. Xi, Y.-H. Kao, M. Yang, and H. J. Chao, “Petabit optical switch for data center networks,” Tech. Rep., Polytechnic Institute of New York University, 2010.

Ye, X.

X. Ye, Y. Yin, S. J. B. Yoo, P. Mejia, R. Proietti, and V. Akella, “DOS: A scalable optical switch for datacenters,” in Proc. 6th ACM/IEEE Symp. on Architectures for Networking and Communications Systems (ANCS), 2010, p. 24.

Yeung, K. L.

B. Wu, K. L. Yeung, M. Hamdi, and X. Li, “Minimizing internal speedup for performance guaranteed switches with optical fabrics,” IEEE/ACM Trans. Netw., vol.  17, no. 2, pp. 632–645, Apr. 2009.
[CrossRef]

B. Wu, K. L. Yeung, P.-H. Ho, and X. H. Jiang, “Minimum delay scheduling for performance guaranteed switches with optical fabrics,” J. Lightwave Technol., vol.  27, no. 16, pp. 3453–3465, Aug. 2009.
[CrossRef]

Yin, Y.

X. Ye, Y. Yin, S. J. B. Yoo, P. Mejia, R. Proietti, and V. Akella, “DOS: A scalable optical switch for datacenters,” in Proc. 6th ACM/IEEE Symp. on Architectures for Networking and Communications Systems (ANCS), 2010, p. 24.

Yoo, S. J. B.

X. Ye, Y. Yin, S. J. B. Yoo, P. Mejia, R. Proietti, and V. Akella, “DOS: A scalable optical switch for datacenters,” in Proc. 6th ACM/IEEE Symp. on Architectures for Networking and Communications Systems (ANCS), 2010, p. 24.

Yuan, J.

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

Yuan, X.

X. Yuan, W. Nienaber, Z. Duan, and R. Melhem, “Oblivious routing in fat-tree based system area networks with uncertain traffic demands,” IEEE/ACM Trans. Netw., vol.  17, no. 5, pp. 1439–1452, 2009.
[CrossRef]

Zhang, D.

Zhang, Q.

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

Zhang, Y.

Y. Zhang and N. Ansari, “On architecture design, congestion notification, TCP incast and power consumption in data centers,” IEEE Commun. Surv. Tutorials, vol.  15, no. 1, pp. 39–64, 2013.
[CrossRef]

C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, “DCell: A scalable and fault-tolerant network structure for data centers,” Comput. Commun. Rev., vol.  38, no. 4, pp. 75–86, Oct. 2008.
[CrossRef]

A. Singla, A. Singh, K. Ramachandran, L. Xu, and Y. Zhang, “Proteus: A topology malleable data center network,” in Proc. Ninth ACM SIGCOMM Workshop on Hot Topics in Networks (Hotnets), 2010, p. 8.

Y. Zhang and N. Ansari, “On mitigating TCP incast in data center networks,” IEEE INFOCOM, Shanghai, 2011, pp. 51–55.

Y. Zhang and N. Ansari, “HERO: Hierarchical energy optimization for data center networks,” IEEE Syst. J., to be published.

Zhani, M.

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

Zhao, X.

C. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag., vol.  48, no. 7, pp. 32–39, July 2010.
[CrossRef]

Zheng, Z.

Z. Zheng, T. Zhou, M. Lyu, and I. King, “Component ranking for fault-tolerant cloud applications,” IEEE Trans. Serv. Comput., vol.  5, no. 4, pp. 540–550, 2012.

Zhou, T.

Z. Zheng, T. Zhou, M. Lyu, and I. King, “Component ranking for fault-tolerant cloud applications,” IEEE Trans. Serv. Comput., vol.  5, no. 4, pp. 540–550, 2012.

Comput. Commun. Rev. (3)

R. N. Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat, “PortLand: A scalable fault-tolerant layer 2 data center network fabric,” Comput. Commun. Rev., vol.  39, no. 4, pp. 39–50, Oct. 2009.
[CrossRef]

C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, “DCell: A scalable and fault-tolerant network structure for data centers,” Comput. Commun. Rev., vol.  38, no. 4, pp. 75–86, Oct. 2008.
[CrossRef]

L. Vaquero, L. Rodero-Merino, J. Caceres, and M. Lindner, “A break in the clouds: Towards a cloud definition,” Comput. Commun. Rev., vol.  39, no. 1, pp. 50–55, Jan. 2009.
[CrossRef]

IEEE Commun. Lett. (1)

F. Wang and M. Hamdi, “Strictly non-blocking conditions for the central-stage buffered Clos-network,” IEEE Commun. Lett., vol.  12, no. 3, pp. 206–208, 2008.
[CrossRef]

IEEE Commun. Mag. (2)

C. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag., vol.  48, no. 7, pp. 32–39, July 2010.
[CrossRef]

H. J. Chao, Z. Jing, and S. Y. Liew, “Matching algorithms for three-stage bufferless Clos network switches,” IEEE Commun. Mag., vol.  41, no. 10, pp. 46–54, 2003.
[CrossRef]

IEEE Commun. Surv. Tutorials (3)

M. Bari, R. Boutaba, R. Esteves, L. Granville, M. Podlesny, M. Rabbani, Q. Zhang, and M. Zhani, “Data center network virtualization: A survey,” IEEE Commun. Surv. Tutorials, vol.  15, no. 2, pp. 909–928, 2013.
[CrossRef]

C. Kachris and I. Tomkos, “A survey on optical interconnects for data centers,” IEEE Commun. Surv. Tutorials, vol.  14, no. 4, pp. 1021–1036, 2012.
[CrossRef]

Y. Zhang and N. Ansari, “On architecture design, congestion notification, TCP incast and power consumption in data centers,” IEEE Commun. Surv. Tutorials, vol.  15, no. 1, pp. 39–64, 2013.
[CrossRef]

IEEE Trans. Comput. (1)

F. O. Sem-Jacobsen, T. Skeie, O. Lysne, and J. Duato, “Dynamic fault tolerance in fat trees,” IEEE Trans. Comput., vol.  60, no. 4, pp. 508–525, 2011.
[CrossRef]

IEEE Trans. Parallel Distrib. Syst. (1)

S. Coll, F. J. Mora, J. Duato, and F. Petrini, “Efficient and scalable hardware-based multicast in fat-tree networks,” IEEE Trans. Parallel Distrib. Syst., vol.  20, no. 9, pp. 1285–1298, 2009.

IEEE Trans. Serv. Comput. (1)

Z. Zheng, T. Zhou, M. Lyu, and I. King, “Component ranking for fault-tolerant cloud applications,” IEEE Trans. Serv. Comput., vol.  5, no. 4, pp. 540–550, 2012.

IEEE/ACM Trans. Netw. (4)

B. Towles and W. J. Dally, “Guaranteed scheduling for switches with configuration overhead,” IEEE/ACM Trans. Netw., vol.  11, no. 5, pp. 835–847, Oct. 2003.
[CrossRef]

K. Chen, C. Guo, H. Wu, J. Yuan, Z. Feng, Y. Chen, S. Lu, and W. Wu, “DAC: Generic and automatic address configuration for data center networks,” IEEE/ACM Trans. Netw., vol.  20, no. 1, pp. 84–99, 2012.
[CrossRef]

B. Wu, K. L. Yeung, M. Hamdi, and X. Li, “Minimizing internal speedup for performance guaranteed switches with optical fabrics,” IEEE/ACM Trans. Netw., vol.  17, no. 2, pp. 632–645, Apr. 2009.
[CrossRef]

X. Yuan, W. Nienaber, Z. Duan, and R. Melhem, “Oblivious routing in fat-tree based system area networks with uncertain traffic demands,” IEEE/ACM Trans. Netw., vol.  17, no. 5, pp. 1439–1452, 2009.
[CrossRef]

J. Lightwave Technol. (3)

Other (15)

Y. Zhang and N. Ansari, “On mitigating TCP incast in data center networks,” IEEE INFOCOM, Shanghai, 2011, pp. 51–55.

P. Wright, T. Harmer, J. Hawkins, and Y. L. Sun, “A commodity-focused multi-cloud marketplace exemplar application,” in 2011 IEEE Int. Conf. on Cloud Computing (CLOUD), 2011, pp. 590–597.

G. Wang, D. G. Andersen, M. Kaminsky, K. Papagiannaki, T. E. Ng, M. Kozuch, and M. Ryan, “c-Through: Part-time optics in data centers,” in Proc. ACM SIGCOMM, 2010, pp. 327–338.

N. Farrington, G. Porter, S. Radhakrishnan, H. H. Bazzaz, V. Subramanya, Y. Fainman, G. Papen, and A. Vahdat, “Helios: A hybrid electrical/optical switch architecture for modular data centers,” in Proc. ACM SIGCOMM, 2010, pp. 339–350.

X. Ye, Y. Yin, S. J. B. Yoo, P. Mejia, R. Proietti, and V. Akella, “DOS: A scalable optical switch for datacenters,” in Proc. 6th ACM/IEEE Symp. on Architectures for Networking and Communications Systems (ANCS), 2010, p. 24.

A. Singla, A. Singh, K. Ramachandran, L. Xu, and Y. Zhang, “Proteus: A topology malleable data center network,” in Proc. Ninth ACM SIGCOMM Workshop on Hot Topics in Networks (Hotnets), 2010, p. 8.

K. Xi, Y.-H. Kao, M. Yang, and H. J. Chao, “Petabit optical switch for data center networks,” Tech. Rep., Polytechnic Institute of New York University, 2010.

J. Gripp, J. E. Simsarian, J. D. LeGrange, P. Bernasconi, and D. T. Neilson, “Photonic terabit routers: The IRIS project,” in Optical Fiber Communication Conf., 2010, paper OThP3.

T. Hoff, “Google architecture,” July2007 [Online]. Available: http://highscalability.com/google-architecture .

J. Snyder, “Microsoft: Datacenter growth defies Moore’s law,” 2007 [Online]. Available: http://www.pcworld.com/article/id,130921/article.html .

U.S. Environmental Protection Agency, “Report to congress on server and data center efficiency (public law 109-431),” ENERGY STAR Program, Aug. 2007.

D. D. Kandlur and T. W. Keller, “Green data centers and hot chips,” in Proc. 46th Annu. Design Automation Conf., 2009, pp. 888–890.

A. Benner, “Optical interconnect opportunities in supercomputers and high end computing,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), 2012, paper OTu2B.4.

“Vision and roadmap: Routing telecom and data centers toward efficient energy use,” poster presented at Workshop on Routing Telecom and Data Centers, 2009.

Y. Zhang and N. Ansari, “HERO: Hierarchical energy optimization for data center networks,” IEEE Syst. J., to be published.

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Figures (9)

Fig. 1.
Fig. 1.

Examples of Google DCNs. (a) Google DCN in Belgium and (b) lakeside site in Dalles, Oregon.

Fig. 2.
Fig. 2.

Distributed placement of optical switches and racks in DCNs.

Fig. 3.
Fig. 3.

Scalability-related cost function.

Fig. 4.
Fig. 4.

Heuristic algorithm DDP for distributed DCN placement. (a) Component set clustering (Stage I of DDP) and (b) component set placement and traffic routing (Stage II of DDP).

Fig. 5.
Fig. 5.

ILP-based optimal solution with a system cost of 238,181 for distributed placement of the DCN. (a) Pan-European COST 239 network, (b) internal traffic load between two DCN server racks, (c) demand at each node and demand routing, (d) link cost in distance (kilometers), (e) simulation parameters, and (f) inter-node DCN internal traffic.

Fig. 6.
Fig. 6.

Heuristic DDP solution with an overall system cost of 271,238 (13.88% above the optimal solution in Fig. 5). (a) Heuristic DDP solution, (b) demand at each node and demand routing, and (c) inter-node DCN internal traffic.

Fig. 7.
Fig. 7.

Component set placement changes with θ.

Fig. 8.
Fig. 8.

Solution changes with the scalability-related cost function in Fig. 3.

Fig. 9.
Fig. 9.

Solution changes with Ps.

Equations (24)

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minimize{uVPsSu+mVkKΔkJmk+uVvVPuvFuv+mVnV,n>muVθ(Pmu+Pnu)TmnuuVP0Yu}.
mVRmi=1,iR;
XmnijRmi+Rnj1,m,nV:m<n,i,jR:ij;
Su1αmVnV,n>mTmnu,uV;
Su1βvVFuv,uV;
Jmk1γ(iRRmiNk),mV,k;
uVTmnu=iRjR,jirijXmnij,m,nV:m<n;
vVFuv+mVnV,n>mTmnuQ,uV;
uVFuv=dv,vV;
YuSu,uV;
YuiRRui,uV.
Ai=Ps+PCiP0QCi.
NC=[1Q(L+vVdv)].
Bu=PC+PI+PDTI+TD.
C1<C2<C3<<Ck1<Ck<.
Pk=PC+PI+kCkDk,
Tk=TI+kDk,
Bk=PkTk,
Pk=Pk1+CkDk,
Tk=Tk1+Dk.
Pk1+CkDkTk1+Dk<Pk1Tk1.
Ck1<Ck<Pk1Tk1=Pk2+Ck1Dk1Tk2+Dk1,
Ck1<Pk2Tk2.
Bk1<Bk2.