Abstract

It has been demonstrated that the overflow priority classification approximation (OPCA) is an accurate method for blocking probability evaluation for various networks and systems, including optical burst switched networks with deflection routing. OPCA is a hierarchical algorithm that requires fixed-point iterations in each layer of its hierarchy. This may imply a long running time. We prove here that the OPCA iterations alternately produce upper and lower bounds that consistently become closer to each other as more fixed-point iterations in each layer are used, and we demonstrate numerically that only a small number of iterations per layer are required for the bounds to be sufficiently close to each other. This behavior is demonstrated for various system parameters including offered load, number of channels per trunk, and maximum allowable number of deflections.

© 2013 Optical Society of America

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Corrections

Shuo Li, Meiqian Wang, Eric W. M. Wong, Vyacheslav Abramov, and Moshe Zukerman, "Bounds of the Overflow Priority Classification for Blocking Probability Approximation in OBS Networks: Errata," J. Opt. Commun. Netw. 5, 1467-1467 (2013)
https://www.osapublishing.org/jocn/abstract.cfm?uri=jocn-5-12-1467

References

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2012 (1)

2010 (1)

S. Bregni, A. Caruso, and A. Pattavina, “Buffering-deflection tradeoffs in optical burst switching,” Photon. Netw. Commun., vol.  20, no. 2, pp. 193–200, Aug. 2010.
[CrossRef]

2009 (2)

2008 (1)

A. Al Amin, M. Takenaka, T. Tanemura, K. Shimizu, R. Inohara, K. Nishimura, M. Usami, Y. Takita, Y. Kai, H. Onaka, H. Uetsuka, and Y. Nakano, “Demonstration of deflection routing with layer-2 evaluation at 40  Gb/s in a three-node optical burst switching testbed,” IEEE Photon. Technol. Lett., vol.  20, no. 3, pp. 178–180, Feb. 2008.
[CrossRef]

2007 (4)

E. W. M. Wong, A. Zalesky, Z. Rosberg, and M. Zukerman, “A new method for approximating blocking probability in overflow loss networks,” Comput. Netw., vol.  51, no. 11, pp. 2958–2975, Aug. 2007.
[CrossRef]

G. Raskutti, A. Zalesky, E. W. M. Wong, and M. Zukerman, “Enhanced blocking probability evaluation method for circuit-switched trunk reservation networks,” IEEE Commun. Lett., vol.  11, no. 6, pp. 543–545, June 2007.
[CrossRef]

A. Zalesky, H. Vu, Z. Rosberg, E. Wong, and M. Zukerman, “OBS contention resolution performance,” Perform. Eval., vol.  64, no. 4, pp. 357–373, 2007.

A. Zalesky, H. L. Vu, Z. Rosberg, E. W. M. Wong, and M. Zukerman, “Stabilizing deflection routing in optical burst switched networks,” IEEE J. Sel. Areas Commun., vol.  25, no. 6, pp. 3–19, Aug. 2007.
[CrossRef]

2006 (1)

Z. Rosberg, A. Zalesky, H. L. Vu, and M. Zukerman, “Analysis of OBS networks with limited wavelength conversion,” IEEE/ACM Trans. Netw., vol.  14, no. 5, pp. 1118–1127, Oct. 2006.
[CrossRef]

2005 (1)

S. Lee, K. Sriram, H. Kim, and J. Song, “Contention-based limited deflection routing protocol in optical burst-switched networks,” IEEE J. Sel. Areas Commun., vol.  23, no. 8, pp. 1596–1611, Aug. 2005.
[CrossRef]

2004 (1)

Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: A new area in optical networking research,” IEEE Network, vol.  18, no. 3, pp. 16–23, May–June 2004.

2003 (4)

T. Battestilli and H. Perros, “An introduction to optical burst switching,” IEEE Commun. Mag., vol.  41, no. 8, pp. S10–S15, Aug. 2003.
[CrossRef]

Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance analyses of optical burst-switching networks,” IEEE J. Sel. Areas Commun., vol.  21, no. 7, pp. 1187–1197, Sept. 2003.
[CrossRef]

S. Yao, B. Mukherjee, S. Yoo, and S. Dixit, “A unified study of contention-resolution schemes in optical packet-switched networks,” J. Lightwave Technol., vol.  21, no. 3, pp. 672–683, Mar. 2003.
[CrossRef]

T. Kuri, H. Toda, and K. Kitayama, “Dense wavelength-division multiplexing millimeter-wave-band radio-on-fiber signal transmission with photonic downconversion,” J. Lightwave Technol., vol.  21, no. 6, pp. 1510–1517, 2003.
[CrossRef]

2002 (2)

2001 (2)

L. Xu, H. Perros, and G. Rouskas, “Techniques for optical packet switching and optical burst switching,” IEEE Commun. Mag., vol.  39, no. 1, pp. 136–142, Jan. 2001.
[CrossRef]

M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next-generation optical Internet,” IEEE Commun. Mag., vol.  39, no. 2, pp. 98–104, Feb. 2001.
[CrossRef]

2000 (2)

S. Verma, H. Chaskar, and R. Ravikanth, “Optical burst switching: A viable solution for terabit IP backbone,” IEEE Network, vol.  14, pp. 48–53, Nov./Dec. 2000.
[CrossRef]

C. Qiao, “Labeled optical burst switching for IP-over-WDM integration,” IEEE Commun. Mag., vol.  38, no. 9, pp. 104–114, Sept. 2000.
[CrossRef]

1999 (1)

C. Qiao and M. Yoo, “Optical burst switching (OBS)—a new paradigm for an optical Internet,” J. High Speed Netw., vol.  8, no. 1, pp. 69–84, Jan. 1999.

1998 (1)

1990 (2)

A. Ghafoor, M. Guizani, and S. Sheikh, “Architecture of an all-optical circuit-switched multistage interconnection network,” IEEE J. Sel. Areas Commun., vol.  8, no. 8, pp. 1595–1607, Oct. 1990.
[CrossRef]

K. P. Krishnan, “The convexity of loss rate in an Erlang loss system and sojourn in an Erlang delay system with respect to arrival and service rates,” IEEE Trans. Commun., vol.  38, no. 9, pp. 1314–1316, Sept. 1990.
[CrossRef]

1986 (1)

F. P. Kelly, “Blocking probabilities in large circuit-switched networks,” Adv. Appl. Probab., vol.  18, no. 2, pp. 473–505, June 1986.

1979 (1)

D. Gale, “The game of Hex and the Brouwer fixed-point theorem,” Am. Math. Monthly, vol.  86, no. 10, pp. 818–827, 1979.

Abe, T.

Y. Mori, T. Abe, H. Pan, Y. Zhu, Y.-B. Choi, and H. Okada, “Effective flow-rate control for the deflection routing based optical burst switching networks,” in Proc. Asia-Pacific Conf. on Communications, Aug. 2006.

Abramov, V.

E. Altman, R. El-Azouzi, and V. Abramov, “Non-cooperative routing in loss networks,” Perform. Eval., vol.  49, no. 1–4, pp. 257–272, 2002.

Al Amin, A.

A. Al Amin, M. Takenaka, T. Tanemura, K. Shimizu, R. Inohara, K. Nishimura, M. Usami, Y. Takita, Y. Kai, H. Onaka, H. Uetsuka, and Y. Nakano, “Demonstration of deflection routing with layer-2 evaluation at 40  Gb/s in a three-node optical burst switching testbed,” IEEE Photon. Technol. Lett., vol.  20, no. 3, pp. 178–180, Feb. 2008.
[CrossRef]

Altman, E.

E. Altman, R. El-Azouzi, and V. Abramov, “Non-cooperative routing in loss networks,” Perform. Eval., vol.  49, no. 1–4, pp. 257–272, 2002.

Aly, W.

M. Levesque, H. Elbiaze, and W. Aly, “Adaptive threshold-based decision for efficient hybrid deflection and retransmission scheme in OBS networks,” in Proc. 13th Int. Conf. Optical Network Design and Modeling, Feb. 2009, pp. 55–60.

Andonovic, I.

Baliga, J.

Barker, K.

K. Barker, A. Benner, R. Hoare, A. Hoisie, A. Jones, D. Kerbyson, D. Li, R. Melhem, R. Rajamony, E. Schenfeld, S. Shao, C. Stunkel, and P. Walker, “On the feasibility of optical circuit switching for high performance computing systems,” in Proc. ACM/IEEE Conf. Supercomputing, Nov. 2005.

Battestilli, T.

T. Battestilli and H. Perros, “An introduction to optical burst switching,” IEEE Commun. Mag., vol.  41, no. 8, pp. S10–S15, Aug. 2003.
[CrossRef]

Bauknecht, R.

Belbekkouche, A.

A. Belbekkouche, A. Hafid, and M. Gendreau, “A reinforcement learning-based deflection routing scheme for buffer-less OBS networks,” in Proc. IEEE GLOBECOM, Nov. 2008.

Benner, A.

K. Barker, A. Benner, R. Hoare, A. Hoisie, A. Jones, D. Kerbyson, D. Li, R. Melhem, R. Rajamony, E. Schenfeld, S. Shao, C. Stunkel, and P. Walker, “On the feasibility of optical circuit switching for high performance computing systems,” in Proc. ACM/IEEE Conf. Supercomputing, Nov. 2005.

Beyranvand, H.

Blumenthal, D.

D. Blumenthal, “Optical packet switching,” in Proc. IEEE Lasers and Electro-Optics Society, vol. 2, Nov. 2004, pp. 910–912.

Bostica, B.

Bregni, S.

S. Bregni, A. Caruso, and A. Pattavina, “Buffering-deflection tradeoffs in optical burst switching,” Photon. Netw. Commun., vol.  20, no. 2, pp. 193–200, Aug. 2010.
[CrossRef]

Burzio, M.

Callegati, F.

Caruso, A.

S. Bregni, A. Caruso, and A. Pattavina, “Buffering-deflection tradeoffs in optical burst switching,” Photon. Netw. Commun., vol.  20, no. 2, pp. 193–200, Aug. 2010.
[CrossRef]

Casoni, M.

Chaskar, H.

S. Verma, H. Chaskar, and R. Ravikanth, “Optical burst switching: A viable solution for terabit IP backbone,” IEEE Network, vol.  14, pp. 48–53, Nov./Dec. 2000.
[CrossRef]

Chen, A.

W.-Y. Yuen, A. Chen, A.-S. Wong, and C.-T. Lea, “Buffer-efficient routing strategies for optical circuit switching networks,” in Proc. BroadNets 2004, Oct. 2004, pp. 328–330.

Chen, Y.

Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: A new area in optical networking research,” IEEE Network, vol.  18, no. 3, pp. 16–23, May–June 2004.

Y. Chen, H. Wu, D. Xu, and C. Qiao, “Performance analysis of optical burst switched node with deflection routing,” in Proc. IEEE Int. Conf. Communications, vol. 2, May 2003, pp. 1355–1359.

Chiaroni, D.

Choi, Y.-B.

Y. Mori, T. Abe, H. Pan, Y. Zhu, Y.-B. Choi, and H. Okada, “Effective flow-rate control for the deflection routing based optical burst switching networks,” in Proc. Asia-Pacific Conf. on Communications, Aug. 2006.

Clerot, F.

Cooper, R. B.

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

Danielsen, S. L.

Detti, A.

Dixit, S.

S. Yao, B. Mukherjee, S. Yoo, and S. Dixit, “A unified study of contention-resolution schemes in optical packet-switched networks,” J. Lightwave Technol., vol.  21, no. 3, pp. 672–683, Mar. 2003.
[CrossRef]

M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next-generation optical Internet,” IEEE Commun. Mag., vol.  39, no. 2, pp. 98–104, Feb. 2001.
[CrossRef]

Dong, W.

W. Dong, M. Fu, Z. Le, and X. Sun, “AIMD control for deflection routing in OBS networks,” in Proc. Asia Communications and Photonics Conf. and Exhibition, Nov. 2009.

Dorgeuille, F.

Dupas, A.

El-Azouzi, R.

E. Altman, R. El-Azouzi, and V. Abramov, “Non-cooperative routing in loss networks,” Perform. Eval., vol.  49, no. 1–4, pp. 257–272, 2002.

Elbiaze, H.

M. Levesque, H. Elbiaze, and W. Aly, “Adaptive threshold-based decision for efficient hybrid deflection and retransmission scheme in OBS networks,” in Proc. 13th Int. Conf. Optical Network Design and Modeling, Feb. 2009, pp. 55–60.

Eramo, V.

Farahmand, F.

T. Ozugur, F. Farahmand, and D. Verchere, “Single-anchored soft bandwidth allocation system with deflection routing for optical burst switching,” in Proc. High Performance Switching and Routing, 2002, pp. 257–261.

Franzen, A.

Fu, M.

W. Dong, M. Fu, Z. Le, and X. Sun, “AIMD control for deflection routing in OBS networks,” in Proc. Asia Communications and Photonics Conf. and Exhibition, Nov. 2009.

Gale, D.

D. Gale, “The game of Hex and the Brouwer fixed-point theorem,” Am. Math. Monthly, vol.  86, no. 10, pp. 818–827, 1979.

Gambini, P.

Gendreau, M.

A. Belbekkouche, A. Hafid, and M. Gendreau, “A reinforcement learning-based deflection routing scheme for buffer-less OBS networks,” in Proc. IEEE GLOBECOM, Nov. 2008.

Ghafoor, A.

A. Ghafoor, M. Guizani, and S. Sheikh, “Architecture of an all-optical circuit-switched multistage interconnection network,” IEEE J. Sel. Areas Commun., vol.  8, no. 8, pp. 1595–1607, Oct. 1990.
[CrossRef]

Golmie, N.

K. Sriram, D. Griffith, S. Lee, and N. Golmie, “Optical burst switching: Benefits and challenges,” in Proc. 1st Int. Workshop on Optical Burst Switching (WOBS)/Opticomm, Dallas, TX, Oct. 2003, pp. 1–12.

Griffith, D.

K. Sriram, D. Griffith, S. Lee, and N. Golmie, “Optical burst switching: Benefits and challenges,” in Proc. 1st Int. Workshop on Optical Burst Switching (WOBS)/Opticomm, Dallas, TX, Oct. 2003, pp. 1–12.

Guillemot, C.

Guizani, M.

A. Ghafoor, M. Guizani, and S. Sheikh, “Architecture of an all-optical circuit-switched multistage interconnection network,” IEEE J. Sel. Areas Commun., vol.  8, no. 8, pp. 1595–1607, Oct. 1990.
[CrossRef]

Hafid, A.

A. Belbekkouche, A. Hafid, and M. Gendreau, “A reinforcement learning-based deflection routing scheme for buffer-less OBS networks,” in Proc. IEEE GLOBECOM, Nov. 2008.

Hansen, P. B.

Hoare, R.

K. Barker, A. Benner, R. Hoare, A. Hoisie, A. Jones, D. Kerbyson, D. Li, R. Melhem, R. Rajamony, E. Schenfeld, S. Shao, C. Stunkel, and P. Walker, “On the feasibility of optical circuit switching for high performance computing systems,” in Proc. ACM/IEEE Conf. Supercomputing, Nov. 2005.

Hoisie, A.

K. Barker, A. Benner, R. Hoare, A. Hoisie, A. Jones, D. Kerbyson, D. Li, R. Melhem, R. Rajamony, E. Schenfeld, S. Shao, C. Stunkel, and P. Walker, “On the feasibility of optical circuit switching for high performance computing systems,” in Proc. ACM/IEEE Conf. Supercomputing, Nov. 2005.

Horiguchi, S.

S.-H. Ngo, X. Jian, and S. Horiguchi, “Hybrid deflection and retransmission routing schemes for OBS networks,” in Proc. High Performance Switching and Routing, 2006.

Hsu, C.-F.

C.-F. Hsu, T.-L. Liu, and N.-F. Huang, “Performance analysis of deflection routing in optical burst-switched networks,” in Proc. IEEE INFOCOM, vol. 1, 2002, pp. 66–73.

Huang, N.-F.

C.-F. Hsu, T.-L. Liu, and N.-F. Huang, “Performance analysis of deflection routing in optical burst-switched networks,” in Proc. IEEE INFOCOM, vol. 1, 2002, pp. 66–73.

Hunter, D. K.

Inohara, R.

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M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next-generation optical Internet,” IEEE Commun. Mag., vol.  39, no. 2, pp. 98–104, Feb. 2001.
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A. Munir, S. Tanwir, and S. Zaidi, “Requests provisioning algorithms for dynamic optical circuit switched (DOCS) networks: A survey,” in Proc. IEEE 13th Int. Multitopic Conf., Dec. 2009.

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E. W. M. Wong, J. Baliga, M. Zukerman, A. Zalesky, and G. Raskutti, “A new method for blocking probability evaluation in OBS/OPS networks with deflection routing,” J. Lightwave Technol., vol.  27, no. 23, pp. 5335–5347, Dec. 2009.
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Y. Mori, T. Abe, H. Pan, Y. Zhu, Y.-B. Choi, and H. Okada, “Effective flow-rate control for the deflection routing based optical burst switching networks,” in Proc. Asia-Pacific Conf. on Communications, Aug. 2006.

Zucchelli, L.

Zukerman, M.

E. W. M. Wong, J. Baliga, M. Zukerman, A. Zalesky, and G. Raskutti, “A new method for blocking probability evaluation in OBS/OPS networks with deflection routing,” J. Lightwave Technol., vol.  27, no. 23, pp. 5335–5347, Dec. 2009.
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M. Zukerman, “Back to the future,” IEEE Commun. Mag., vol.  47, no. 11, pp. 36–38, 2009.
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G. Raskutti, A. Zalesky, E. W. M. Wong, and M. Zukerman, “Enhanced blocking probability evaluation method for circuit-switched trunk reservation networks,” IEEE Commun. Lett., vol.  11, no. 6, pp. 543–545, June 2007.
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Model for LBT: (a) a network model and (b) an example for direct trunks and LBT in the network shown in (a).

Fig. 2.
Fig. 2.

NSF network topology in which each solid line represents two unidirectional trunks in opposing directions.

Fig. 3.
Fig. 3.

Blocking probabilities in the NSFNET.

Fig. 4.
Fig. 4.

Bounds of OPCA blocking probabilities in the NSFNET with a different offered load to each directional SD pair.

Fig. 5.
Fig. 5.

Bounds of OPCA blocking probabilities in the NSFNET with a different number of channels per trunk (C), in which the offered load to each directional SD pair is 0.4C.

Fig. 6.
Fig. 6.

Bounds of OPCA blocking probabilities in the NSFNET with a different maximum allowable number of deflections (D), in which the offered load to each directional SD pair is 20 erlangs.

Fig. 7.
Fig. 7.

Positions of Q(x,a1) and Q(x,a2).

Tables (3)

Tables Icon

TABLE I Ingress and Egress SD Pairs

Tables Icon

TABLE II Comparison of the Times Used by the EFPA and OPCA to Calculate the Blocking Probabilities in NSFNET

Tables Icon

TABLE III Comparison of the Times Used by the EFPA and OPCA in Each Layer to Calculate the Blocking Probabilities in NSFNET With 10,000 Channels per Trunk

Equations (81)

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

{Um(0),Um,j1(1),Um,j2(1),,Um,jn(Tm)}
T(m)=min{Tm,D}
ai20(m)=ai10(m)(1bi10)=ρm(1bi10).
al1k+1(m)=aik(m)bik,
al2k+2(m)=al1k+1(m)bl1k+1=aik(m)bikbl1k+1.
ajk=mβajk(m).
I(i,j,Um,p(k))={1,if i,jE and trunk i strictly precedes(not necessarily immediately) trunk jalong k deflection route Um,p(k)0,otherwise.
ajk=mβ,jUm,p(k)ρm,pkiE(1I(i,j,Um,p(k))bik),
aj0=mβ,jUm(0)ρmiE(1I(i,j,Um(0))bi0)
a˜jk=h=0kajh.
b¯jk=E(a˜jk,Cj),
bjk={b¯j0,k=0b¯jka˜jkb¯jk1a˜jk1ajk1kD.
aj0U=mβ,jUm(0)ρmiE(1I(i,j,Um(0))bi0L),
aj0L=mβ,jUm(0)ρmiE(1I(i,j,Um(0))bi0U),
bj0L=E(aj0L,Cj),
bj0U=E(aj0U,Cj).
aj0L(m)=ρmiE(1I(i,j,Um(0))bi0U),
aj0U(m)=ρmiE(1I(i,j,Um(0))bi0L).
ajkU=mβ,qE,jUm,q(k){aqk1U(m)bqk1U×iE(1I(i,j,Um,q(k))bikL)},
ajkL=mβ,qE,jUm,q(k){aqk1L(m)bqk1L×iE(1I(i,j,Um,q(k))bikU)}.
bjk=E(a˜jk1+ajk,Cj)(a˜jk1+ajk)E(a˜jk1,Cj)(a˜jk1)ajk.
E(a˜jk1L+ajkL,Cj)(a˜jk1L+ajkL)E(a˜jk1L,Cj)(a˜jk1L)ajkLE(a˜jk1U+ajkL,Cj)(a˜jk1U+ajkL)E(a˜jk1U,Cj)(a˜jk1U)ajkLE(a˜jk1U+ajkU,Cj)(a˜jk1U+ajkU)E(a˜jk1U,Cj)(a˜jk1U)ajkU,E(a˜jk1L+ajkL,Cj)(a˜jk1L+ajkL)E(a˜jk1L,Cj)(a˜jk1L)ajkLE(a˜jk1L+ajkU,Cj)(a˜jk1L+ajkU)E(a˜jk1L,Cj)(a˜jk1L)ajkUE(a˜jk1U+ajkU,Cj)(a˜jk1U+ajkU)E(a˜jk1U,Cj)(a˜jk1U)ajkU.
bjkU=E(a˜jk1U+ajkU,Cj)(a˜jk1U+ajkU)E(a˜jk1U,Cj)(a˜jk1U)ajkU,
bjkL=E(a˜jk1L+ajkL,Cj)(a˜jk1L+ajkL)E(a˜jk1L,Cj)(a˜jk1L)ajkL.
ajkL(m)=qE,jUm,q(k)aqk1L(m)bqk1L×iE(1I(i,j,Um,q(k))bikU),
ajkU(m)=qE,jUm,q(k)aqk1U(m)bqk1U×iE(1I(i,j,Um,q(k))bikL).
ajk2z1>ajk2z+1>ajkmax*,ajk2z2<ajk2z<ajkmin*,bjk2z2>bjk2z>bjkmax*,bjk2z1<bjk2z+1<bjkmin*.
aj01=mβ,jUm(0)ρmiE(1I(i,j,Um(0))(1H(0,i))bi01)×(1I(i,j,Um(0))H(0,i)bi0L),
I(i,j,Um(0))={1,if i,jE and trunk i strictly precedes(not necessarily immediately) trunk jalong primary route of SD pair m0,otherwise,
H(k,i)={1,if trunk i is a LBT in layer k0,if trunk i is a direct trunk in layer k.
bj02=E(aj01,Cj).
aj02=mβ,jUm(0)ρmiE(1I(i,j,Um(0))(1H(0,i))bi02)×(1I(i,j,Um(0))H(0,i)bi0U).
bj03=E(aj02,Cj).
aj03=mβ,jUm(0)ρmiE(1I(i,j,Um(0))(1H(0,i))bi03)×(1I(i,j,Um(0))H(0,i)bi0L),
bj04=E(aj03,Cj).
bj0s=E(aj0s1,Cj),
aj0s=mβ,jUm(0)ρmiE(1I(i,j,Um(0))(1H(0,i))bi0s)×(1I(i,j,Um(0))(1H(0,i))bi0L.
aj01>aj03>aj05>>limzaj02z1=aj0max*,
aj02<aj04<aj06<<limzaj02z=aj0min*.
bj02>bj04>bj06>>limzbj02z=bj0max*,
bj01<bj03<bj05<<limzbj02z1=bj0min*.
aj02i1>aj02i+1>aj02i+2>aj02i,
aj02i+1>aj02i+3>aj02i+4>aj02i+2
bj02i>bj02i+2>bj02i+3>bj02i+1.
aj02i+1>aj02i+3>aj02i+2>aj02i.
bj02i+2>bj02i+4>bj02i+3>bj02i+1.
aj1=mβ,qE,jUm,q(1)aq0(m)bq0iE(1I(i,j,Um,q(1))×(1H(1,i))bi1)(1I(i,j,Um,q(1))H(1,i)bi1).
aj1U=mβ,qE,jUm,q(1)aq0U(m)bq0UiE(1I(i,j,Um,q(1))×(1H(1,i))bi1)(1I(i,j,Um,q(1))H(1,i)biL),
aj1L=mβ,qE,jUm,q(1)aq0L(m)bq0LiE(1I(i,j,Um,q(1))×(1H(1,i))bi1)(1I(i,j,Um,q(1))H(1,i)biU).
aj12z1=mβ,qE,jUm,q(1)aq0U(m)bq0UiE{1I(i,j,Um,q(1))×(1H(1,i))bi12z1}{1I(i,j,Um,q(1))H(1,i)bi1L},
aj12z=mβ,qE,jUm,q(1)aq0L(m)bq0LiE{1I(i,j,Um,q(1))×(1H(1,i))bi12z}{1I(i,j,Um,q(1))H(1,i)bi1U}, bj12z=E(aj0U+aj12z1,Cj)(aj0U+aj12z1)E(aj0U,Cj)(aj0U)aj12z1,bj12z+1=E(aj0L+aj12z,Cj)(aj0L+aj12z)E(aj0L,Cj)(aj0L)aj12z.
aj11>aj13>aj15>>limzaj12z1aj1max*,
aj12<aj14<aj16<<limzaj12zaj1min*,
bj12>bj14>bj16>>limzbj12zbj1max*,
bj11<bj13<bj15<<limzbj12z1bj1min*.
aj1L(m)=qE,jUm,q(1)aq0L(m)bq0LiE(1I(i,j,Um,q(1))bi1U),
aj1U(m)=qE,jUm,q(1)aq0U(m)bq0UiE(1I(i,j,Um,q(1))bi1L).
ajk2z1=mβ,qE,jUm,q(k)aq(k1)U(m)bq(k1)U×iE{1I(i,j,Um,q(k))(1H(k,i))bik2z1}×{1I(i,j,Um,q(k))H(k,i)bikL},
ajk2z=mβ,qE,jUm,q(k)aq(k1)L(m)bq(k1)L×iE{1I(i,j,Um,q(k))(1H(k,i))bik2z}×{1I(i,j,Um,q(k))H(k,i)bikU},
bjk2z=E(a˜j(k1)U+ajk2z1,Cj)(a˜j(k1)U+ajk2z1)ajk2z1E(a˜j(k1)U,Cj)(a˜j(k1)U)ajk2z1,
bjk2z+1=E(a˜j(k1)L+ajk2z,Cj)(a˜j(k1)L+ajk2z)ajk2zE(a˜j(k1)L,Cj)(a˜j(k1)L)ajk2z.
ajk1>ajk3>ajk5>>limzajk2z1ajkmax*,
ajk2<ajk4<ajk6<<limzajk2zajkmin*,
bjk2>bjk4>bjk6>>limzbjk2zbjkmax*,
bjk1<bjk3<bjk5<<limzbjk2z1bjkmin*.
ajkL(m)=qE,jUm,q(k){aqk1L(m)bqk1LiE(1I(i,j,Um,q(k))bikU)},
ajkU(m)=qE,jUm,q(k){aqk1U(m)bqk1UiE(1I(i,j,Um,q(k))bikL)}.
aji+12z1>aji+12z1>aji+12z>aji+12z,bji+12z>bji+12z>bji+12z1>bji+12z1.
BU(m)=1ρmiUm(0)(1bi0U)ρmqEk=1T(m)ρUm,q(k)LpUm,q(k)(1bpkU)ρm,
ρUm,q(k)L={aq(k1)L(m)bq(k1)L,if path Um,q(h) exists,0,otherwise,
BL(m)=1ρmiUm(0)(1bi0L)ρmqEh=1T(m)ρUm,q(k)UpUm,q(k)(1bpkL)ρm,
ρUm,q(k)U={aq(k1)U(m)bq(k1)U,if path Um,q(k) exists,0,otherwise.
B(m)=1ρmiUm(0)(1bi0)ρmqEk=1T(m)ρUm,q(h)pUm,q(k)(1bpk)ρm,
ρUm,q(k)={aqk1(m)bqk1,if path Um,q(k) exists0,otherwise.
BUBmax*Bmin*BL.
BU=mβBU(m)#SD,
BL=mβBL(m)#SD,
BUBmax*Bmin*BL.
Q(x,a)=F(x+a)F(x)a
Q(x,a)=F(x+a)F(x)a=1axx+aF(y)dy.
Q(x,a)=F(x+θa),