Abstract

Hybrid optical architectures combining path and packet switching can be good candidates for future optical networks because they exploit the best of both worlds. However, the optimization of some parameters of the hybrid switch by some metrics is vital to maximize the benefit of the hybrid architecture. Blocking rate and reservation delay are two of the most important performance metrics in the path switching layer of the hybrid architecture. In this paper, we propose an analytical method with an improved backward blocking analysis to compute both blocking rates and reservation delays in path switching optical WDM networks with destination-initiated reservation allowing retrial of failed reservation attempts. On a mesh topology, we show that the results of our analytical method and simulations were close to each other, while the analytical method was several orders of magnitude faster than the simulation, which may allow much faster performance analysis, design, and optimization of hybrid networks.

© 2013 Optical Society of America

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    [CrossRef]

2006

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

S. Bjornstad, D. Hjelme, and N. Stol, “A packet-switched hybrid optical network with service guarantees,” IEEE J. Sel. Areas Commun., vol.  24, no. 8, pp. 97–107, Aug. 2006.
[CrossRef]

2005

X. Zheng, M. Veeraraghavan, N. S. V. Rao, Q. Wu, and M. Zhu, “CHEETAH: Circuit-switched high-speed end-to-end transport architecture testbed,” IEEE Commun. Mag., vol.  43, no. 8, pp. S11–S17, Aug. 2005.
[CrossRef]

K. Lu, G. Xiao, and I. Chlamtac, “Analysis of blocking probability for distributed lightpath establishment in WDM optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 1, pp. 187–197, Feb. 2005.
[CrossRef]

2004

A. Sridharan and K. N. Sivarajan, “Blocking in all-optical networks,” IEEE/ACM Trans. Netw., vol.  12, no. 2, pp. 384–397, Apr. 2004.
[CrossRef]

2003

2000

Y. Zhu, G. N. Rouskas, and H. G. Perros, “A path decomposition approach for computing blocking probabilities in wavelength-routing networks,” IEEE/ACM Trans. Netw., vol.  8, no. 6, pp. 747–762, Dec. 2000.
[CrossRef]

1999

X. Yuan, R. Melhem, R. Gupta, Y. Mei, and C. Qiao, “Distributed control protocols for wavelength reservation and their performance evaluation,” Photon. Netw. Commun., vol.  1, no. 3, pp. 207–218, 1999.
[CrossRef]

1996

A. Birman, “Computing approximate blocking probabilities for a class of all-optical networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 852–857, June 1996.
[CrossRef]

M. Kovacevic and A. S. Acampora, “Benefits of wavelength translation in all-optical clear-channel networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 868–880, June 1996.
[CrossRef]

R. Ramaswami and K. N. Sivarajan, “Design of logical topologies for wavelength-routed optical networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 840–851, June 1996.
[CrossRef]

1993

S. P. Chung, A. Kashper, and K. W. Ross, “Computing approximate blocking probability for large loss networks with state-dependent routing,” IEEE/ACM Trans. Netw., vol.  1, no. 1, pp. 105–115, Feb. 1993.
[CrossRef]

1992

I. Chlamtac, A. Ganz, and G. Karmi, “Lightpath communications: A novel approach to high bandwidth optical WANs,” IEEE Trans. Commun., vol.  40, no. 7, pp. 1171–1182, July 1992.
[CrossRef]

Acampora, A. S.

M. Kovacevic and A. S. Acampora, “Benefits of wavelength translation in all-optical clear-channel networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 868–880, June 1996.
[CrossRef]

Alparslan, O.

O. Alparslan, S. Arakawa, and M. Murata, “Computing path blocking probabilities for traffic splitting in optical hybrid switching networks,” in Proc. IEEE Int. Conf. Communications, Ottawa, Canada, June 2012.

Arakawa, S.

O. Alparslan, S. Arakawa, and M. Murata, “Computing path blocking probabilities for traffic splitting in optical hybrid switching networks,” in Proc. IEEE Int. Conf. Communications, Ottawa, Canada, June 2012.

S. Arakawa, K. Miyamoto, M. Murata, and H. Miyahara, “Performance analyses of wavelength reservation methods for high-speed data transfer in photonic networks,” in Proc. ITC-CSCC‘99, San Antonio, TX, July 1999, pp. 828–831.

S. Arakawa, N. Tsutsui, and M. Murata, “A biologically-inspired wavelength resource allocation for optical path/packet integrated networks,” in Proc. 15th Conf. Optical Network Design and Modeling, Feb. 2011.

Berger, L.

L. Berger, “Generalized multi-protocol label switching (GMPLS) signaling resource reservation protocol-traffic engineering (RSVP-TE) extensions,” IETF RFC 3473, Jan. 2003.

Birman, A.

A. Birman, “Computing approximate blocking probabilities for a class of all-optical networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 852–857, June 1996.
[CrossRef]

Bjornstad, S.

S. Bjornstad, D. Hjelme, and N. Stol, “A packet-switched hybrid optical network with service guarantees,” IEEE J. Sel. Areas Commun., vol.  24, no. 8, pp. 97–107, Aug. 2006.
[CrossRef]

Cao, J.

J. Cao, W. S. Cleveland, D. Lin, and D. X. Sun, “On the nonstationarity of Internet traffic,” in Proc. SIGMETRICS/Performance’01, 2001, pp. 102–112.

Cheyns, J.

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

Chlamtac, I.

K. Lu, G. Xiao, and I. Chlamtac, “Analysis of blocking probability for distributed lightpath establishment in WDM optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 1, pp. 187–197, Feb. 2005.
[CrossRef]

I. Chlamtac, A. Ganz, and G. Karmi, “Lightpath communications: A novel approach to high bandwidth optical WANs,” IEEE Trans. Commun., vol.  40, no. 7, pp. 1171–1182, July 1992.
[CrossRef]

Chung, S. P.

S. P. Chung, A. Kashper, and K. W. Ross, “Computing approximate blocking probability for large loss networks with state-dependent routing,” IEEE/ACM Trans. Netw., vol.  1, no. 1, pp. 105–115, Feb. 1993.
[CrossRef]

Cleveland, W. S.

J. Cao, W. S. Cleveland, D. Lin, and D. X. Sun, “On the nonstationarity of Internet traffic,” in Proc. SIGMETRICS/Performance’01, 2001, pp. 102–112.

Colle, D.

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

Cooper, R. B.

R. B. Cooper and S. Katz, “Analysis of alternative routing networks with account taken of nonrandomness of overflow traffic,” Bell Telephone Lab. Tech. Rep., 1964.

De Turck, F.

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

De Winter, D.

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

Demeester, P.

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

Ganz, A.

I. Chlamtac, A. Ganz, and G. Karmi, “Lightpath communications: A novel approach to high bandwidth optical WANs,” IEEE Trans. Commun., vol.  40, no. 7, pp. 1171–1182, July 1992.
[CrossRef]

Gupta, R.

X. Yuan, R. Melhem, R. Gupta, Y. Mei, and C. Qiao, “Distributed control protocols for wavelength reservation and their performance evaluation,” Photon. Netw. Commun., vol.  1, no. 3, pp. 207–218, 1999.
[CrossRef]

Harai, H.

H. Harai, “Optical packet and circuit integrated network system and testbed,” invited talk presented at the ECOC 2010 Co-located Workshop, Torino, Italy, Sept. 2010 [Online]. Available: http://www.ecoc2010.org/contents/attached/c20/WS_2_Harai.pdf .

Hjelme, D.

S. Bjornstad, D. Hjelme, and N. Stol, “A packet-switched hybrid optical network with service guarantees,” IEEE J. Sel. Areas Commun., vol.  24, no. 8, pp. 97–107, Aug. 2006.
[CrossRef]

Horiuchi, Y.

F. Xue, S. Yoo, H. Yokoyama, and Y. Horiuchi, “Performance analysis of wavelength-routed optical networks with connection request retrials,” in Proc. IEEE Int. Conf. Communications, May 2005, pp. 1813–1818.

Jue, J. P.

J. P. Jue and G. Xiao, “Analysis of blocking probability for connection management schemes in optical networks,” in Proc. IEEE GLOBECOM, San Antonio, TX, Nov. 2001, pp. 1546–1550.

Karmi, G.

I. Chlamtac, A. Ganz, and G. Karmi, “Lightpath communications: A novel approach to high bandwidth optical WANs,” IEEE Trans. Commun., vol.  40, no. 7, pp. 1171–1182, July 1992.
[CrossRef]

Kashper, A.

S. P. Chung, A. Kashper, and K. W. Ross, “Computing approximate blocking probability for large loss networks with state-dependent routing,” IEEE/ACM Trans. Netw., vol.  1, no. 1, pp. 105–115, Feb. 1993.
[CrossRef]

Katz, S.

R. B. Cooper and S. Katz, “Analysis of alternative routing networks with account taken of nonrandomness of overflow traffic,” Bell Telephone Lab. Tech. Rep., 1964.

Kovacevic, M.

M. Kovacevic and A. S. Acampora, “Benefits of wavelength translation in all-optical clear-channel networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 868–880, June 1996.
[CrossRef]

Lin, D.

J. Cao, W. S. Cleveland, D. Lin, and D. X. Sun, “On the nonstationarity of Internet traffic,” in Proc. SIGMETRICS/Performance’01, 2001, pp. 102–112.

Lu, K.

K. Lu, G. Xiao, and I. Chlamtac, “Analysis of blocking probability for distributed lightpath establishment in WDM optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 1, pp. 187–197, Feb. 2005.
[CrossRef]

Mei, Y.

X. Yuan, R. Melhem, R. Gupta, Y. Mei, and C. Qiao, “Distributed control protocols for wavelength reservation and their performance evaluation,” Photon. Netw. Commun., vol.  1, no. 3, pp. 207–218, 1999.
[CrossRef]

Melhem, R.

X. Yuan, R. Melhem, R. Gupta, Y. Mei, and C. Qiao, “Distributed control protocols for wavelength reservation and their performance evaluation,” Photon. Netw. Commun., vol.  1, no. 3, pp. 207–218, 1999.
[CrossRef]

Miyahara, H.

S. Arakawa, K. Miyamoto, M. Murata, and H. Miyahara, “Performance analyses of wavelength reservation methods for high-speed data transfer in photonic networks,” in Proc. ITC-CSCC‘99, San Antonio, TX, July 1999, pp. 828–831.

Miyamoto, K.

S. Arakawa, K. Miyamoto, M. Murata, and H. Miyahara, “Performance analyses of wavelength reservation methods for high-speed data transfer in photonic networks,” in Proc. ITC-CSCC‘99, San Antonio, TX, July 1999, pp. 828–831.

Murata, M.

S. Arakawa, K. Miyamoto, M. Murata, and H. Miyahara, “Performance analyses of wavelength reservation methods for high-speed data transfer in photonic networks,” in Proc. ITC-CSCC‘99, San Antonio, TX, July 1999, pp. 828–831.

S. Arakawa, N. Tsutsui, and M. Murata, “A biologically-inspired wavelength resource allocation for optical path/packet integrated networks,” in Proc. 15th Conf. Optical Network Design and Modeling, Feb. 2011.

O. Alparslan, S. Arakawa, and M. Murata, “Computing path blocking probabilities for traffic splitting in optical hybrid switching networks,” in Proc. IEEE Int. Conf. Communications, Ottawa, Canada, June 2012.

Papadimitriou, G.

Papazoglou, C.

Perros, H. G.

Y. Zhu, G. N. Rouskas, and H. G. Perros, “A path decomposition approach for computing blocking probabilities in wavelength-routing networks,” IEEE/ACM Trans. Netw., vol.  8, no. 6, pp. 747–762, Dec. 2000.
[CrossRef]

Pickavet, M.

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

Pomportsis, A.

Qiao, C.

X. Yuan, R. Melhem, R. Gupta, Y. Mei, and C. Qiao, “Distributed control protocols for wavelength reservation and their performance evaluation,” Photon. Netw. Commun., vol.  1, no. 3, pp. 207–218, 1999.
[CrossRef]

Ramaswami, R.

R. Ramaswami and K. N. Sivarajan, “Design of logical topologies for wavelength-routed optical networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 840–851, June 1996.
[CrossRef]

Rao, N. S. V.

X. Zheng, M. Veeraraghavan, N. S. V. Rao, Q. Wu, and M. Zhu, “CHEETAH: Circuit-switched high-speed end-to-end transport architecture testbed,” IEEE Commun. Mag., vol.  43, no. 8, pp. S11–S17, Aug. 2005.
[CrossRef]

Ross, K. W.

S. P. Chung, A. Kashper, and K. W. Ross, “Computing approximate blocking probability for large loss networks with state-dependent routing,” IEEE/ACM Trans. Netw., vol.  1, no. 1, pp. 105–115, Feb. 1993.
[CrossRef]

Rouskas, G. N.

Y. Zhu, G. N. Rouskas, and H. G. Perros, “A path decomposition approach for computing blocking probabilities in wavelength-routing networks,” IEEE/ACM Trans. Netw., vol.  8, no. 6, pp. 747–762, Dec. 2000.
[CrossRef]

Sivarajan, K. N.

A. Sridharan and K. N. Sivarajan, “Blocking in all-optical networks,” IEEE/ACM Trans. Netw., vol.  12, no. 2, pp. 384–397, Apr. 2004.
[CrossRef]

R. Ramaswami and K. N. Sivarajan, “Design of logical topologies for wavelength-routed optical networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 840–851, June 1996.
[CrossRef]

Sridharan, A.

A. Sridharan and K. N. Sivarajan, “Blocking in all-optical networks,” IEEE/ACM Trans. Netw., vol.  12, no. 2, pp. 384–397, Apr. 2004.
[CrossRef]

Stol, N.

S. Bjornstad, D. Hjelme, and N. Stol, “A packet-switched hybrid optical network with service guarantees,” IEEE J. Sel. Areas Commun., vol.  24, no. 8, pp. 97–107, Aug. 2006.
[CrossRef]

Sun, D. X.

J. Cao, W. S. Cleveland, D. Lin, and D. X. Sun, “On the nonstationarity of Internet traffic,” in Proc. SIGMETRICS/Performance’01, 2001, pp. 102–112.

Tsutsui, N.

S. Arakawa, N. Tsutsui, and M. Murata, “A biologically-inspired wavelength resource allocation for optical path/packet integrated networks,” in Proc. 15th Conf. Optical Network Design and Modeling, Feb. 2011.

Van Breusegern, E.

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

Veeraraghavan, M.

X. Zheng, M. Veeraraghavan, N. S. V. Rao, Q. Wu, and M. Zhu, “CHEETAH: Circuit-switched high-speed end-to-end transport architecture testbed,” IEEE Commun. Mag., vol.  43, no. 8, pp. S11–S17, Aug. 2005.
[CrossRef]

Wu, Q.

X. Zheng, M. Veeraraghavan, N. S. V. Rao, Q. Wu, and M. Zhu, “CHEETAH: Circuit-switched high-speed end-to-end transport architecture testbed,” IEEE Commun. Mag., vol.  43, no. 8, pp. S11–S17, Aug. 2005.
[CrossRef]

Xiao, G.

K. Lu, G. Xiao, and I. Chlamtac, “Analysis of blocking probability for distributed lightpath establishment in WDM optical networks,” IEEE/ACM Trans. Netw., vol.  13, no. 1, pp. 187–197, Feb. 2005.
[CrossRef]

J. P. Jue and G. Xiao, “Analysis of blocking probability for connection management schemes in optical networks,” in Proc. IEEE GLOBECOM, San Antonio, TX, Nov. 2001, pp. 1546–1550.

Xue, F.

F. Xue, S. Yoo, H. Yokoyama, and Y. Horiuchi, “Performance analysis of wavelength-routed optical networks with connection request retrials,” in Proc. IEEE Int. Conf. Communications, May 2005, pp. 1813–1818.

Yokoyama, H.

F. Xue, S. Yoo, H. Yokoyama, and Y. Horiuchi, “Performance analysis of wavelength-routed optical networks with connection request retrials,” in Proc. IEEE Int. Conf. Communications, May 2005, pp. 1813–1818.

Yoo, S.

F. Xue, S. Yoo, H. Yokoyama, and Y. Horiuchi, “Performance analysis of wavelength-routed optical networks with connection request retrials,” in Proc. IEEE Int. Conf. Communications, May 2005, pp. 1813–1818.

Yuan, X.

X. Yuan, R. Melhem, R. Gupta, Y. Mei, and C. Qiao, “Distributed control protocols for wavelength reservation and their performance evaluation,” Photon. Netw. Commun., vol.  1, no. 3, pp. 207–218, 1999.
[CrossRef]

Zheng, X.

X. Zheng, M. Veeraraghavan, N. S. V. Rao, Q. Wu, and M. Zhu, “CHEETAH: Circuit-switched high-speed end-to-end transport architecture testbed,” IEEE Commun. Mag., vol.  43, no. 8, pp. S11–S17, Aug. 2005.
[CrossRef]

Zhu, M.

X. Zheng, M. Veeraraghavan, N. S. V. Rao, Q. Wu, and M. Zhu, “CHEETAH: Circuit-switched high-speed end-to-end transport architecture testbed,” IEEE Commun. Mag., vol.  43, no. 8, pp. S11–S17, Aug. 2005.
[CrossRef]

Zhu, Y.

Y. Zhu, G. N. Rouskas, and H. G. Perros, “A path decomposition approach for computing blocking probabilities in wavelength-routing networks,” IEEE/ACM Trans. Netw., vol.  8, no. 6, pp. 747–762, Dec. 2000.
[CrossRef]

IEEE Commun. Mag.

X. Zheng, M. Veeraraghavan, N. S. V. Rao, Q. Wu, and M. Zhu, “CHEETAH: Circuit-switched high-speed end-to-end transport architecture testbed,” IEEE Commun. Mag., vol.  43, no. 8, pp. S11–S17, Aug. 2005.
[CrossRef]

IEEE J. Sel. Areas Commun.

E. Van Breusegern, J. Cheyns, D. De Winter, D. Colle, M. Pickavet, F. De Turck, and P. Demeester, “Overspill routing in optical networks: A true hybrid optical network design,” IEEE J. Sel. Areas Commun., vol.  24, no. 4, pp. 13–25, 2006.
[CrossRef]

S. Bjornstad, D. Hjelme, and N. Stol, “A packet-switched hybrid optical network with service guarantees,” IEEE J. Sel. Areas Commun., vol.  24, no. 8, pp. 97–107, Aug. 2006.
[CrossRef]

A. Birman, “Computing approximate blocking probabilities for a class of all-optical networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 852–857, June 1996.
[CrossRef]

M. Kovacevic and A. S. Acampora, “Benefits of wavelength translation in all-optical clear-channel networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 868–880, June 1996.
[CrossRef]

R. Ramaswami and K. N. Sivarajan, “Design of logical topologies for wavelength-routed optical networks,” IEEE J. Sel. Areas Commun., vol.  14, no. 5, pp. 840–851, June 1996.
[CrossRef]

IEEE Trans. Commun.

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Figures (9)

Fig. 1.
Fig. 1.

Birth–death process.

Fig. 2.
Fig. 2.

EON topology.

Fig. 3.
Fig. 3.

Blocking rate when the request arrival rate was 30flows/s and the hop delay was 10 ms.

Fig. 4.
Fig. 4.

Blocking rate when the request arrival rate was 500flows/s and the hop delay was 10 ms.

Fig. 5.
Fig. 5.

Blocking rate when the request arrival rate was 500flows/s and the hop delay was 1 ms.

Fig. 6.
Fig. 6.

Blocking rate with retrial when the total reservation request arrival rate was 500flows/s and the hop delay was 10 ms.

Fig. 7.
Fig. 7.

Wavelength utilization rate when the hop delay was 10 ms.

Fig. 8.
Fig. 8.

Blocking rate versus load in (a) EON, (b) NSFNET, (c) five-node ring, and (d) 11-node ring topologies.

Fig. 9.
Fig. 9.

Reservation delay of flows, which successfully reserve a path, when the total reservation request arrival rate is (a) 30flows/s and (b) 500flows/s.

Tables (1)

Tables Icon

TABLE I EON Traffic Matrix

Equations (25)

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

pm,kn=[αm,0nαm,1nαm,k1nk!(μmn)k]pm,0n
pm,0n=11+z=1W1z!j=0z1αm,jnμmn,
qm,h2=i=0Wj=0WR(Wh|Wi,Wj)pm,i1pm,j2,
R(h|i,j)=(ih)(Wijh)(Wj),
qm,hn=i=0Wj=0WR(Wh|Wi,Wj)qm,in1pm,jn,
λm,k2=emh=0W1i=0WR(Wh|Wi,Wk)pm,i1=em(1i=0WR(0|Wi,Wk)pm,i1).
λm,kn=em(1i=0WR(0|Wi,Wk)qm,in1).
γmd=emh=0W1qm,hd=em(1qm,Wd),
Λm,kn=mM,mn=mn,n=d(m)λm,kn+mM,mm,mn=mn,mn+cmn+c,nd(m)γm,kn
γm,kn1=γmneΛm,kn1(dn+1)D/(Wk),
γmn1=j=0W1pm,jn1γm,jn1=γmnj=0W1pm,jn1eΛm,jn1(dn+1)D/(Wj).
δm=γm1=γmdx=2dj=0W1pm,jx1eΛm,jx1(dx+1)D/(Wj).
βmn=γmnδm.
smn=nD+φ.
tmn={nDifn20otherwise.
μmn=mM,mn=mnγmnmM,mn=mn(δmsmn+βmntmn),
αm,kn=mM,mn=mn,n=d(m)λm,kn+mM,mn=mn,nd(m)γm,kn,
Lm=1δmem=1(1qm,Wd)x=2dj=0W1pm,jx1eΛm,jx1(dx+1)D/(Wj).
eml=emn=1l(rLm)n1.
Lml=1(1Lm)n=1l(rLm)n1.
Tm1=Rm1+φ=dD+φ.
Nm=qm,W1+n=2d1n(qm,Wnqm,Wn1)+d(Lmqm,Wd1)Lm,
Amn=dD+(n1)(NmD+B),
Pmn=(1Lm)(rLm)n1.
Rml=n=1lAmnPmn1Lml=n=1l(dD+(n1)(NmD+B))(rLm)n1n=1l(rLm)n1,