Abstract

This paper focuses on the energy consumption minimization problem of an optical transport network extending over a very wide area. In particular, the specific problem of power-aware routing and wavelength assignment (PA-RWA) is addressed considering a transparent multi-fiber optical network. An integer linear programming formulation is provided for the static lightpath establishment problem and a few heuristics are proposed to solve the problem in the case of dynamic lightpath establishment. To solve the routing problem we propose a novel algorithm named load based cost; it is based on a cost function that considers the load of each optical fiber to compute link weights. We also propose two new wavelength assignment algorithms: the first one is derived from the first fit heuristic whilst the second one is based on an innovative approach. It works like a least cost routing algorithm assigning a cost to each wavelength for each link in the path and then minimizes the total cost. Our proposal is compared with a number of already known RWA algorithms showing that it is able to reduce the power consumption of the network by about 20–30%, depending on the amount of traffic treated, with respect to state of the art algorithms with a difference of about 10% with respect to the lower bound. In order to provide a complete analysis of the feasibility of the proposed heuristics, the blocking probability is evaluated: the results show that, unlike the other power-aware algorithms here considered, our proposal shows a low impairment with respect to the least congested path–first fit alternative. Another important result regards the better performance of the proposed wavelength assignment algorithm with respect to first fit.

© 2011 OSA

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    [CrossRef]
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2011 (1)

A. Coiro, M. Listanti, A. Valenti, and F. Matera, "Reducing power consumption in wavelength routed networks by selective switch off of optical links," J. Sel. Top. Quantum Electron. 17, (2), 428‒436 (2011).
[CrossRef]

2010 (1)

A. Silvestri, A. Valenti, S. Pompei, F. Matera, A. Cianfrani, and A. Coiro, "Energy saving in optical transport networks exploiting transmission properties and wavelength path optimization," Opt. Switching Netw. 7, (3), 108‒114 (2010).
[CrossRef]

2009 (3)

2006 (1)

2005 (1)

M. Yano, F. Yamagishi, and T. Tsuda, "Optical MEMS for photonic switching—compact and stable optical crossconnect switches for simple, fast, and flexible wavelength applications in recent photonic networks," IEEE J. Sel. Top. Quantum Electron. 11, (2), 383‒394 (2005).
[CrossRef]

2004 (1)

K. J. Christensen, C. Gunaratne, B. Nordman, and A. D. George, "The next frontier for communications networks: power management," Comput. Commun. 27, 1758‒1770 (2004).
[CrossRef]

2000 (1)

H. Zang, J. Jue, and B. Mukherjee, "A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks," Opt. Networks Mag. 14, (1), 47‒60 (2000).

Aleksic, S.

Audenaert, P.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

Ayre, R.

J. Baliga, R. Ayre, W. V. Sorin, K. Hinton, and R. S. Tucker, "Energy consumption in optical IP networks," J. Lightwave Technol. 27, (13), 2391‒2403 (2009).
[CrossRef]

J. Baliga, R. Ayre, K. Hinton, and R. S. Tucker, "Photonic switching and the energy bottleneck," Proc. Photon. Switching, Aug. 2007, San Francisco, pp. 125‒126.

Baliga, J.

J. Baliga, R. Ayre, W. V. Sorin, K. Hinton, and R. S. Tucker, "Energy consumption in optical IP networks," J. Lightwave Technol. 27, (13), 2391‒2403 (2009).
[CrossRef]

J. Baliga, R. Ayre, K. Hinton, and R. S. Tucker, "Photonic switching and the energy bottleneck," Proc. Photon. Switching, Aug. 2007, San Francisco, pp. 125‒126.

Barry, R. A.

S. Subramaniam and R. A. Barry, "Wavelength assignment in fixed routing WDM networks," Proc. ICC ’97, Vol. 1, June 1997, Montreal, Canada, pp. 406‒410.

Chan, K.

K. Chan and T. P. Yum, "Analysis of least congested path routing in WDM lightwave networks," Proc. IEEE INFOCOM ’94, Vol. 2, Apr. 1994, Toronto, Canada, pp. 962‒969.

Chiaraviglio, L.

Y. Wu, L. Chiaraviglio, M. Mellia, and F. Neri, "Power-aware routing and wavelength assignment in optical networks," European Conf. Optical Communications (ECOC) 2009, Sept. 2009, Wien, pp. 1‒2.

L. Chiaraviglio, M. Mellia, and F. Neri, "Reducing power consumption in backbone networks," IEEE ICC 09, June 2009, Dresden, pp. 1‒6.

L. Chiaraviglio, M. Mellia, and F. Neri, "Energy-aware backbone networks: a case study," IEEE ICC 09 – Comm. Workshops, June 2009, Dresden, pp. 1‒5.

Christensen, K. J.

K. J. Christensen, C. Gunaratne, B. Nordman, and A. D. George, "The next frontier for communications networks: power management," Comput. Commun. 27, 1758‒1770 (2004).
[CrossRef]

Cianfrani, A.

A. Silvestri, A. Valenti, S. Pompei, F. Matera, A. Cianfrani, and A. Coiro, "Energy saving in optical transport networks exploiting transmission properties and wavelength path optimization," Opt. Switching Netw. 7, (3), 108‒114 (2010).
[CrossRef]

Coiro, A.

A. Coiro, M. Listanti, A. Valenti, and F. Matera, "Reducing power consumption in wavelength routed networks by selective switch off of optical links," J. Sel. Top. Quantum Electron. 17, (2), 428‒436 (2011).
[CrossRef]

A. Silvestri, A. Valenti, S. Pompei, F. Matera, A. Cianfrani, and A. Coiro, "Energy saving in optical transport networks exploiting transmission properties and wavelength path optimization," Opt. Switching Netw. 7, (3), 108‒114 (2010).
[CrossRef]

Colle, D.

W. V. Heddeghem, M. De Groote, W. Vereecken, D. Colle, M. Pickavet, and P. Demeester, "Energy-efficiency in telecommunication networks: link-by-link versus end-to-end grooming," Proc. ONDM 2010, Feb. 2010, Kyoto (Japan), pp. 1‒6.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

De Groote, M.

W. V. Heddeghem, M. De Groote, W. Vereecken, D. Colle, M. Pickavet, and P. Demeester, "Energy-efficiency in telecommunication networks: link-by-link versus end-to-end grooming," Proc. ONDM 2010, Feb. 2010, Kyoto (Japan), pp. 1‒6.

Demeester, P.

W. V. Heddeghem, M. De Groote, W. Vereecken, D. Colle, M. Pickavet, and P. Demeester, "Energy-efficiency in telecommunication networks: link-by-link versus end-to-end grooming," Proc. ONDM 2010, Feb. 2010, Kyoto (Japan), pp. 1‒6.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

Demeyer, S.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

Develder, C.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

Dhoedt, B.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

George, A. D.

K. J. Christensen, C. Gunaratne, B. Nordman, and A. D. George, "The next frontier for communications networks: power management," Comput. Commun. 27, 1758‒1770 (2004).
[CrossRef]

Gerlach, C.

C. Lange, D. Kosiankowski, C. Gerlach, F. Westphal, and A. Gladisch, "Energy consumption of telecommunication networks," Proc. ECOC 2009, Sept. 2009, Vienna, Austria.

Gladisch, A.

C. Lange, D. Kosiankowski, C. Gerlach, F. Westphal, and A. Gladisch, "Energy consumption of telecommunication networks," Proc. ECOC 2009, Sept. 2009, Vienna, Austria.

Gunaratne, C.

K. J. Christensen, C. Gunaratne, B. Nordman, and A. D. George, "The next frontier for communications networks: power management," Comput. Commun. 27, 1758‒1770 (2004).
[CrossRef]

Gupta, M.

M. Gupta and S. Singh, "Greening of the Internet," Proc. SIGCOMM’03: Applications, Technologies, Architectures, and Protocols for Computer Communications, 2003, Karlsruhe, pp. 19‒26.

Harai, H.

H. Harai, M. Murata, and H. Miyahara, "Performance of alternate routing methods in all-optical switching networks," Proc. IEEE INFOCOM ’97, Vol. 2, Apr. 1997, Kobe, Japan, pp. 516‒524.

Heddeghem, W. V.

W. V. Heddeghem, M. De Groote, W. Vereecken, D. Colle, M. Pickavet, and P. Demeester, "Energy-efficiency in telecommunication networks: link-by-link versus end-to-end grooming," Proc. ONDM 2010, Feb. 2010, Kyoto (Japan), pp. 1‒6.

Hinton, K.

J. Baliga, R. Ayre, W. V. Sorin, K. Hinton, and R. S. Tucker, "Energy consumption in optical IP networks," J. Lightwave Technol. 27, (13), 2391‒2403 (2009).
[CrossRef]

J. Baliga, R. Ayre, K. Hinton, and R. S. Tucker, "Photonic switching and the energy bottleneck," Proc. Photon. Switching, Aug. 2007, San Francisco, pp. 125‒126.

Jue, J.

H. Zang, J. Jue, and B. Mukherjee, "A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks," Opt. Networks Mag. 14, (1), 47‒60 (2000).

Kosiankowski, D.

C. Lange, D. Kosiankowski, C. Gerlach, F. Westphal, and A. Gladisch, "Energy consumption of telecommunication networks," Proc. ECOC 2009, Sept. 2009, Vienna, Austria.

Lange, C.

C. Lange, D. Kosiankowski, C. Gerlach, F. Westphal, and A. Gladisch, "Energy consumption of telecommunication networks," Proc. ECOC 2009, Sept. 2009, Vienna, Austria.

Listanti, M.

A. Coiro, M. Listanti, A. Valenti, and F. Matera, "Reducing power consumption in wavelength routed networks by selective switch off of optical links," J. Sel. Top. Quantum Electron. 17, (2), 428‒436 (2011).
[CrossRef]

Matera, F.

A. Coiro, M. Listanti, A. Valenti, and F. Matera, "Reducing power consumption in wavelength routed networks by selective switch off of optical links," J. Sel. Top. Quantum Electron. 17, (2), 428‒436 (2011).
[CrossRef]

A. Silvestri, A. Valenti, S. Pompei, F. Matera, A. Cianfrani, and A. Coiro, "Energy saving in optical transport networks exploiting transmission properties and wavelength path optimization," Opt. Switching Netw. 7, (3), 108‒114 (2010).
[CrossRef]

Mellia, M.

L. Chiaraviglio, M. Mellia, and F. Neri, "Energy-aware backbone networks: a case study," IEEE ICC 09 – Comm. Workshops, June 2009, Dresden, pp. 1‒5.

L. Chiaraviglio, M. Mellia, and F. Neri, "Reducing power consumption in backbone networks," IEEE ICC 09, June 2009, Dresden, pp. 1‒6.

Y. Wu, L. Chiaraviglio, M. Mellia, and F. Neri, "Power-aware routing and wavelength assignment in optical networks," European Conf. Optical Communications (ECOC) 2009, Sept. 2009, Wien, pp. 1‒2.

Miyahara, H.

H. Harai, M. Murata, and H. Miyahara, "Performance of alternate routing methods in all-optical switching networks," Proc. IEEE INFOCOM ’97, Vol. 2, Apr. 1997, Kobe, Japan, pp. 516‒524.

Mukherjee, B.

H. Zang, J. Jue, and B. Mukherjee, "A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks," Opt. Networks Mag. 14, (1), 47‒60 (2000).

S. Ramamurthy and B. Mukherjee, "Fixed-alternate routing and wavelength conversion in wavelength-routed optical networks," Proc. IEEE GLOBECOM’98, Vol. 4, Nov. 1998, pp. 2295‒2302.

Murakami, M.

M. Murakami and K. Oda, "Power consumption analysis of optical cross-connect equipment for future large capacity optical networks," Proc. ICTON 2009, July 2009, Azores Islands, pp. 1‒4.

Murata, M.

H. Harai, M. Murata, and H. Miyahara, "Performance of alternate routing methods in all-optical switching networks," Proc. IEEE INFOCOM ’97, Vol. 2, Apr. 1997, Kobe, Japan, pp. 516‒524.

Neri, F.

Y. Wu, L. Chiaraviglio, M. Mellia, and F. Neri, "Power-aware routing and wavelength assignment in optical networks," European Conf. Optical Communications (ECOC) 2009, Sept. 2009, Wien, pp. 1‒2.

L. Chiaraviglio, M. Mellia, and F. Neri, "Reducing power consumption in backbone networks," IEEE ICC 09, June 2009, Dresden, pp. 1‒6.

L. Chiaraviglio, M. Mellia, and F. Neri, "Energy-aware backbone networks: a case study," IEEE ICC 09 – Comm. Workshops, June 2009, Dresden, pp. 1‒5.

Nordman, B.

K. J. Christensen, C. Gunaratne, B. Nordman, and A. D. George, "The next frontier for communications networks: power management," Comput. Commun. 27, 1758‒1770 (2004).
[CrossRef]

Oda, K.

M. Murakami and K. Oda, "Power consumption analysis of optical cross-connect equipment for future large capacity optical networks," Proc. ICTON 2009, July 2009, Azores Islands, pp. 1‒4.

Pickavet, M.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

W. V. Heddeghem, M. De Groote, W. Vereecken, D. Colle, M. Pickavet, and P. Demeester, "Energy-efficiency in telecommunication networks: link-by-link versus end-to-end grooming," Proc. ONDM 2010, Feb. 2010, Kyoto (Japan), pp. 1‒6.

Pompei, S.

A. Silvestri, A. Valenti, S. Pompei, F. Matera, A. Cianfrani, and A. Coiro, "Energy saving in optical transport networks exploiting transmission properties and wavelength path optimization," Opt. Switching Netw. 7, (3), 108‒114 (2010).
[CrossRef]

Qiao, C.

X. Zhang and C. Qiao, "Wavelength assignment for dynamic traffic in multi-fibre WDM networks," Proc. 7th Int. Conf. Computer Communications and Networks, Oct. 1998, Lafayette, LA, pp. 479‒485.

Ramamurthy, S.

S. Ramamurthy and B. Mukherjee, "Fixed-alternate routing and wavelength conversion in wavelength-routed optical networks," Proc. IEEE GLOBECOM’98, Vol. 4, Nov. 1998, pp. 2295‒2302.

Shen, G.

Silvestri, A.

A. Silvestri, A. Valenti, S. Pompei, F. Matera, A. Cianfrani, and A. Coiro, "Energy saving in optical transport networks exploiting transmission properties and wavelength path optimization," Opt. Switching Netw. 7, (3), 108‒114 (2010).
[CrossRef]

Singh, S.

M. Gupta and S. Singh, "Greening of the Internet," Proc. SIGCOMM’03: Applications, Technologies, Architectures, and Protocols for Computer Communications, 2003, Karlsruhe, pp. 19‒26.

Sorin, W. V.

Subramaniam, S.

S. Subramaniam and R. A. Barry, "Wavelength assignment in fixed routing WDM networks," Proc. ICC ’97, Vol. 1, June 1997, Montreal, Canada, pp. 406‒410.

Tsuda, T.

M. Yano, F. Yamagishi, and T. Tsuda, "Optical MEMS for photonic switching—compact and stable optical crossconnect switches for simple, fast, and flexible wavelength applications in recent photonic networks," IEEE J. Sel. Top. Quantum Electron. 11, (2), 383‒394 (2005).
[CrossRef]

Tucker, R. S.

Valenti, A.

A. Coiro, M. Listanti, A. Valenti, and F. Matera, "Reducing power consumption in wavelength routed networks by selective switch off of optical links," J. Sel. Top. Quantum Electron. 17, (2), 428‒436 (2011).
[CrossRef]

A. Silvestri, A. Valenti, S. Pompei, F. Matera, A. Cianfrani, and A. Coiro, "Energy saving in optical transport networks exploiting transmission properties and wavelength path optimization," Opt. Switching Netw. 7, (3), 108‒114 (2010).
[CrossRef]

Vereecken, W.

W. V. Heddeghem, M. De Groote, W. Vereecken, D. Colle, M. Pickavet, and P. Demeester, "Energy-efficiency in telecommunication networks: link-by-link versus end-to-end grooming," Proc. ONDM 2010, Feb. 2010, Kyoto (Japan), pp. 1‒6.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

Vermeulen, B.

M. Pickavet, W. Vereecken, S. Demeyer, P. Audenaert, B. Vermeulen, C. Develder, D. Colle, B. Dhoedt, and P. Demeester, "Worldwide energy needs for ICT: the rise of power-aware networking," Proc. IEEE ANTS 2008, Dec. 2008, Bombay, pp. 1‒3.

Westphal, F.

C. Lange, D. Kosiankowski, C. Gerlach, F. Westphal, and A. Gladisch, "Energy consumption of telecommunication networks," Proc. ECOC 2009, Sept. 2009, Vienna, Austria.

Wu, Y.

Y. Wu, L. Chiaraviglio, M. Mellia, and F. Neri, "Power-aware routing and wavelength assignment in optical networks," European Conf. Optical Communications (ECOC) 2009, Sept. 2009, Wien, pp. 1‒2.

Yamagishi, F.

M. Yano, F. Yamagishi, and T. Tsuda, "Optical MEMS for photonic switching—compact and stable optical crossconnect switches for simple, fast, and flexible wavelength applications in recent photonic networks," IEEE J. Sel. Top. Quantum Electron. 11, (2), 383‒394 (2005).
[CrossRef]

Yano, M.

M. Yano, F. Yamagishi, and T. Tsuda, "Optical MEMS for photonic switching—compact and stable optical crossconnect switches for simple, fast, and flexible wavelength applications in recent photonic networks," IEEE J. Sel. Top. Quantum Electron. 11, (2), 383‒394 (2005).
[CrossRef]

Yum, T. P.

K. Chan and T. P. Yum, "Analysis of least congested path routing in WDM lightwave networks," Proc. IEEE INFOCOM ’94, Vol. 2, Apr. 1994, Toronto, Canada, pp. 962‒969.

Zang, H.

H. Zang, J. Jue, and B. Mukherjee, "A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks," Opt. Networks Mag. 14, (1), 47‒60 (2000).

Zhang, X.

X. Zhang and C. Qiao, "Wavelength assignment for dynamic traffic in multi-fibre WDM networks," Proc. 7th Int. Conf. Computer Communications and Networks, Oct. 1998, Lafayette, LA, pp. 479‒485.

Comput. Commun. (1)

K. J. Christensen, C. Gunaratne, B. Nordman, and A. D. George, "The next frontier for communications networks: power management," Comput. Commun. 27, 1758‒1770 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Yano, F. Yamagishi, and T. Tsuda, "Optical MEMS for photonic switching—compact and stable optical crossconnect switches for simple, fast, and flexible wavelength applications in recent photonic networks," IEEE J. Sel. Top. Quantum Electron. 11, (2), 383‒394 (2005).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Commun. Netw. (2)

J. Sel. Top. Quantum Electron. (1)

A. Coiro, M. Listanti, A. Valenti, and F. Matera, "Reducing power consumption in wavelength routed networks by selective switch off of optical links," J. Sel. Top. Quantum Electron. 17, (2), 428‒436 (2011).
[CrossRef]

Opt. Networks Mag. (1)

H. Zang, J. Jue, and B. Mukherjee, "A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks," Opt. Networks Mag. 14, (1), 47‒60 (2000).

Opt. Switching Netw. (1)

A. Silvestri, A. Valenti, S. Pompei, F. Matera, A. Cianfrani, and A. Coiro, "Energy saving in optical transport networks exploiting transmission properties and wavelength path optimization," Opt. Switching Netw. 7, (3), 108‒114 (2010).
[CrossRef]

Other (19)

Y. Wu, L. Chiaraviglio, M. Mellia, and F. Neri, "Power-aware routing and wavelength assignment in optical networks," European Conf. Optical Communications (ECOC) 2009, Sept. 2009, Wien, pp. 1‒2.

ITU Telecommunication Standardization Sector, "Architecture of optical transport network," ITU-T Rec. G.872, Nov. 2001.

http://www.mrv.com/datasheets/LD/PDF300/MRV-LD-OAT_A4_HI.pdf

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Figures (11)

Fig. 1
Fig. 1

(Color online) Network model and devices.

Fig. 2
Fig. 2

(Color online) V-like functions for γ = 1 , γ = 2 , and γ = 4 ; W = 20 is assumed.

Fig. 3
Fig. 3

(Color online) Different cost functions used for fiber cost calculation in the LCW algorithm; W = 10 is assumed.

Fig. 4
Fig. 4

An example of a randomly generated physical network topology.

Fig. 5
Fig. 5

(Color online) Power consumption of different network devices for the LCP | FF RWA algorithm.

Fig. 6
Fig. 6

(Color online) Routing algorithm performance comparison for TF–FF, LAP–FF, and LCW wavelength assignment algorithms.

Fig. 7
Fig. 7

(Color online) Distribution of the number of used fibers for each link, for all routing algorithms; LAP–FF used as the WA algorithm.

Fig. 8
Fig. 8

(Color online) Routing algorithm performance comparison versus traffic load, for LAP–FF, TP–FF, and LCW wavelength assignment algorithms.

Fig. 9
Fig. 9

(Color online) Comparison of path length distributions for all routing algorithms, for low (0.3 Erlang), medium (3 Erlang), and high (20 Erlang) traffic loads. LAP–FF is used as the wavelength assignment algorithm.

Fig. 10
Fig. 10

(Color online) Wavelength assignment algorithm performance comparison versus traffic load, for LCP, ShP, LBC, and MUP routing algorithms.

Fig. 11
Fig. 11

(Color online) Blocking probability versus traffic load for all routing and wavelength assignment algorithms. N = 12 ; W = 40 ; D = 0 . 3 ; fixed number of fibers F = 5 .

Tables (2)

Tables Icon

Table I Power Consumption of Network Devices

Tables Icon

Table II Values of Power Consumption of the Optimal Solution

Equations (23)

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P i Node ( N i , I i , W ) = P ECS + I i P Trans + ( N i W + I i ) P MEMS .
P i , j Fiber ( L i , j ) = L i , j / d amp P ILA + ( P Pre + P Post ) .
Minimize :
P tot = i , j , k ( N i , j P I L A + P E ) x i , j , k .
Subject to :
λ ( f i , j , k s , d , λ ) M x i , j , k ; i , j , s , d V , k K i , j .
s , d ( f i , j , k s , d , λ ) 1 ; i , j V , λ Λ , k K i , j .
i , k ( f i , j , k s , d , λ f j , i , k s , d , λ ) = f s , d , λ ; if  j = s f s , d , λ ; if  j = d   0 ; if  j s d j , s , d V , λ Λ .
λ ( f s , d , λ ) = T s , d ; s , d V .
λ ( f i , j , k s , d , λ ) W ; i , j , s , d V , k K i , j .
k ( x i , j , k ) F i , j ; i , j V .
LC i , j ( t ) = PC i , j LF i , j ( t ) .
LF i , j ( t ) = FS ( LF i , j 1 ( t ) , LF i , j 2 ( t ) , , LF i , j k ( t ) , , LF i , j K ( t ) ) .
LF i , j k ( t ) = FCA ( L i , j k ( t ) ) .
LF i , j k ( t ) = | 2 L i , j k ( t ) / W 1 | γ sign ( 2 L i , j k ( t ) / W 1 ) .
FC i , j k ( t ) = PC i , j LF i , j k ( t ) .
LF i , j k ( t ) = FCA ( L i , j k ( t ) ) .
WC i , j λ = PC i , j FS ( LF i , j 1 ( t ) , , LF i , j k ( t ) , , LF i , j K ( t ) ) .
LF i , j k ( t ) = 1 / ( L i , j k ( t ) + 1 ) .
P a r a l l e l ( x 1 , x 2 , , x N ) = 1 / ( 1 / x 1 + 1 / x 2 + + 1 / x N ) .
Prob { f s , d Avg > 0 } = 1 / 2 ,
Prob { x 1 < f s , d Avg < x 2 | f s , d Avg > 0 } = ( x 2 x 1 ) / Max_f s , d Avg ;
x 1 > 0 ; x 2 < Max _ f s , d Avg .