Abstract

Optical buffering via fiber delay lines is used for contention resolution in optical packet and optical burst switching nodes. This article addresses the problem of exactly finding the blocking probabilities in an asynchronous single-wavelength optical buffer. Packet lengths are assumed to be variable and modeled by phase-type distributions, whereas the packet arrival process is modeled by a Markovian arrival process that can capture autocorrelations in interarrival times. The exact solution is based on the theory of feedback fluid queues for which we propose numerically efficient and stable algorithms. We not only find the packet blocking probabilities but also the entire distribution of the unfinished work in this system from which all performance measures of interest can be derived.

© 2009 Optical Society of America

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  4. I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
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  5. T. Zhang, K. Lu, J. Jue, “Shared fiber delay line buffers in asynchronous optical packet switches,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 118–127, 2006.
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  8. W. Rogiest, K. Laevens, D. Fiems, H. Bruneel, “A performance model for an asynchronous optical buffer,” Perform. Eval., vol. 62, no. 1–4, pp. 313–330, 2005.
    [CrossRef]
  9. W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
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2009 (2)

W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
[CrossRef]

J. Liu, T. T. Lee, X. Jiang, S. Horiguchi, “Blocking and delay analysis of single wavelength optical buffer with general packet size distribution,” J. Lightwave Technol., vol. 27, no. 8, pp. 955–966, 2009.
[CrossRef]

2008 (2)

2006 (1)

T. Zhang, K. Lu, J. Jue, “Shared fiber delay line buffers in asynchronous optical packet switches,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 118–127, 2006.
[CrossRef]

2005 (2)

R. C. Almeida, J. Pelegrini, H. Waldman, “A generic-traffic optical buffer modeling for asynchronous optical switching networks,” IEEE Commun. Lett., vol. 9, no. 2, pp. 175–177, Feb. 2005.
[CrossRef]

W. Rogiest, K. Laevens, D. Fiems, H. Bruneel, “A performance model for an asynchronous optical buffer,” Perform. Eval., vol. 62, no. 1–4, pp. 313–330, 2005.
[CrossRef]

2003 (1)

M. Mandjes, D. Mitra, W. Scheinhardt, “Models of network access using feedback fluid queues,” Queueing Syst., vol. 44, no. 4, pp. 2989–3002, 2003.
[CrossRef]

2000 (1)

F. Callegati, “Optical buffers for variable length packets,” IEEE Commun. Lett., vol. 4, no. 9, pp. 292–294, Sept. 2000.
[CrossRef]

1999 (1)

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

1998 (2)

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

I. Adan, E. van Doorn, J. Resing, W. Scheinhardt, “Analysis of a single-server queue interacting with a fluid reservoir,” Queueing Syst., vol. 29, pp. 313–336, 1998.
[CrossRef]

1996 (3)

S. Asmussen, O. Nerman, M. Olsson, “Fitting phase-type distributions via the EM algorithm,” Scand. J. Stat., vol. 23, pp. 419–441, 1996.

R. Barry, P. Humblet, “Models of blocking probability in all-optical networks with and without wavelength changers,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 858–867, June 1996.
[CrossRef]

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

1995 (1)

V. Paxson, S. Floyd, “Wide-area traffic: the failure of Poisson modeling,” IEEE/ACM Trans. Netw., vol. 3, pp. 226–244, 1995.
[CrossRef]

1991 (1)

D. M. Lucantoni, “New results for the single server queue with a batch Markovian arrival process,” Stoch. Models, vol. 7, pp. 1–46, 1991.
[CrossRef]

1988 (1)

R. Tucker, “Accurate method for analysis of a packet speech multiplexer with limited delay,” IEEE Trans. Commun., vol. 36, no. 4, pp. 479–483, 1988.
[CrossRef]

1982 (1)

D. Anick, D. Mitra, M. M. Sondhi, “Stochastic theory of a data handling system with multiple sources,” Bell Syst. Tech. J., vol. 61, pp. 1871–1894, 1982.
[CrossRef]

Adan, I.

I. Adan, E. van Doorn, J. Resing, W. Scheinhardt, “Analysis of a single-server queue interacting with a fluid reservoir,” Queueing Syst., vol. 29, pp. 313–336, 1998.
[CrossRef]

Akar, N.

H. E. Kankaya, N. Akar, “Solving multi-regime feedback fluid queues,” Stoch. Models, vol. 24, no. 3, pp. 425–450, July 2008.
[CrossRef]

Almeida, R. C.

R. C. Almeida, J. Pelegrini, H. Waldman, “A generic-traffic optical buffer modeling for asynchronous optical switching networks,” IEEE Commun. Lett., vol. 9, no. 2, pp. 175–177, Feb. 2005.
[CrossRef]

Andonovic, I.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Anick, D.

D. Anick, D. Mitra, M. M. Sondhi, “Stochastic theory of a data handling system with multiple sources,” Bell Syst. Tech. J., vol. 61, pp. 1871–1894, 1982.
[CrossRef]

Asmussen, S.

S. Asmussen, O. Nerman, M. Olsson, “Fitting phase-type distributions via the EM algorithm,” Scand. J. Stat., vol. 23, pp. 419–441, 1996.

Barry, R.

R. Barry, P. Humblet, “Models of blocking probability in all-optical networks with and without wavelength changers,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 858–867, June 1996.
[CrossRef]

Blondia, C.

W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
[CrossRef]

Bostica, B.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Bruneel, H.

W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
[CrossRef]

W. Rogiest, K. Laevens, D. Fiems, H. Bruneel, “A performance model for an asynchronous optical buffer,” Perform. Eval., vol. 62, no. 1–4, pp. 313–330, 2005.
[CrossRef]

Callegati, F.

F. Callegati, “Optical buffers for variable length packets,” IEEE Commun. Lett., vol. 4, no. 9, pp. 292–294, Sept. 2000.
[CrossRef]

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Cerisola, M.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Chiaroni, D.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Chlamtac, I.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Choudhury, A.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Cigno, R. L.

L. Muscariello, M. Mellia, M. Meo, R. L. Cigno, M. A. Marsan, “An MMPP-based hierarchical model of Internet traffic,” in 2004 IEEE Int. Conf. Communications, vol. 4, June 2004, pp. 2143–2147.
[CrossRef]

Corazza, G.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Danielsen, S. L.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Fiems, D.

W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
[CrossRef]

W. Rogiest, K. Laevens, D. Fiems, H. Bruneel, “A performance model for an asynchronous optical buffer,” Perform. Eval., vol. 62, no. 1–4, pp. 313–330, 2005.
[CrossRef]

Floyd, S.

V. Paxson, S. Floyd, “Wide-area traffic: the failure of Poisson modeling,” IEEE/ACM Trans. Netw., vol. 3, pp. 226–244, 1995.
[CrossRef]

Fong, T.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Fumagalli, A.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Gambini, P.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins U. Press, 1996.

Gravey, P.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Guillemot, C.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Hansen, P. B.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Haverkort, B. R.

B. R. Haverkort, Performance of Computer and Communication Systems: A Model-based Approach. Wiley, 1998.
[CrossRef]

Henry, M.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Hofmeister, R.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Horiguchi, S.

Houdt, B. V.

W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
[CrossRef]

Humblet, P.

R. Barry, P. Humblet, “Models of blocking probability in all-optical networks with and without wavelength changers,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 858–867, June 1996.
[CrossRef]

Janz, C.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Jiang, X.

Jue, J.

T. Zhang, K. Lu, J. Jue, “Shared fiber delay line buffers in asynchronous optical packet switches,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 118–127, 2006.
[CrossRef]

Kang, S.

S. Kang, D. Sung, “Two-state MMPP modeling of ATM superposed traffic streams based on the characterization of correlated interarrival times,” in IEEE Global Telecommunications Conf., 1995. GLOBECOM ’95, vol. 2, Nov. 1995, pp. 1422–1426.

Kankaya, H. E.

H. E. Kankaya, N. Akar, “Solving multi-regime feedback fluid queues,” Stoch. Models, vol. 24, no. 3, pp. 425–450, July 2008.
[CrossRef]

Kazovsky, L.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Kloch, A.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Krahenbuhl, R.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Laevens, K.

W. Rogiest, K. Laevens, D. Fiems, H. Bruneel, “A performance model for an asynchronous optical buffer,” Perform. Eval., vol. 62, no. 1–4, pp. 313–330, 2005.
[CrossRef]

Lambert, J.

W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
[CrossRef]

Lee, T. T.

Liang, Z.

Liu, J.

Lu, C.-L.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Lu, K.

T. Zhang, K. Lu, J. Jue, “Shared fiber delay line buffers in asynchronous optical packet switches,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 118–127, 2006.
[CrossRef]

Lucantoni, D. M.

D. M. Lucantoni, “New results for the single server queue with a batch Markovian arrival process,” Stoch. Models, vol. 7, pp. 1–46, 1991.
[CrossRef]

Mandjes, M.

M. Mandjes, D. Mitra, W. Scheinhardt, “Models of network access using feedback fluid queues,” Queueing Syst., vol. 44, no. 4, pp. 2989–3002, 2003.
[CrossRef]

Marsan, M. A.

L. Muscariello, M. Mellia, M. Meo, R. L. Cigno, M. A. Marsan, “An MMPP-based hierarchical model of Internet traffic,” in 2004 IEEE Int. Conf. Communications, vol. 4, June 2004, pp. 2143–2147.
[CrossRef]

Mekkittikul, A.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Mellia, M.

L. Muscariello, M. Mellia, M. Meo, R. L. Cigno, M. A. Marsan, “An MMPP-based hierarchical model of Internet traffic,” in 2004 IEEE Int. Conf. Communications, vol. 4, June 2004, pp. 2143–2147.
[CrossRef]

Melman, P.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Meo, M.

L. Muscariello, M. Mellia, M. Meo, R. L. Cigno, M. A. Marsan, “An MMPP-based hierarchical model of Internet traffic,” in 2004 IEEE Int. Conf. Communications, vol. 4, June 2004, pp. 2143–2147.
[CrossRef]

Mitra, D.

M. Mandjes, D. Mitra, W. Scheinhardt, “Models of network access using feedback fluid queues,” Queueing Syst., vol. 44, no. 4, pp. 2989–3002, 2003.
[CrossRef]

D. Anick, D. Mitra, M. M. Sondhi, “Stochastic theory of a data handling system with multiple sources,” Bell Syst. Tech. J., vol. 61, pp. 1871–1894, 1982.
[CrossRef]

Muscariello, L.

L. Muscariello, M. Mellia, M. Meo, R. L. Cigno, M. A. Marsan, “An MMPP-based hierarchical model of Internet traffic,” in 2004 IEEE Int. Conf. Communications, vol. 4, June 2004, pp. 2143–2147.
[CrossRef]

Nelson, W.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Nerman, O.

S. Asmussen, O. Nerman, M. Olsson, “Fitting phase-type distributions via the EM algorithm,” Scand. J. Stat., vol. 23, pp. 419–441, 1996.

Neuts, M. F.

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models. Baltimore, MD: Johns Hopkins U. Press, 1981.

M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications. New York: Marcel Dekker, 1989.

Olsson, M.

S. Asmussen, O. Nerman, M. Olsson, “Fitting phase-type distributions via the EM algorithm,” Scand. J. Stat., vol. 23, pp. 419–441, 1996.

Paxson, V.

V. Paxson, S. Floyd, “Wide-area traffic: the failure of Poisson modeling,” IEEE/ACM Trans. Netw., vol. 3, pp. 226–244, 1995.
[CrossRef]

Pelegrini, J.

R. C. Almeida, J. Pelegrini, H. Waldman, “A generic-traffic optical buffer modeling for asynchronous optical switching networks,” IEEE Commun. Lett., vol. 9, no. 2, pp. 175–177, Feb. 2005.
[CrossRef]

Poggiolini, P.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Qiao, C.

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

Raffaelli, C.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Renaud, M.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Resing, J.

I. Adan, E. van Doorn, J. Resing, W. Scheinhardt, “Analysis of a single-server queue interacting with a fluid reservoir,” Queueing Syst., vol. 29, pp. 313–336, 1998.
[CrossRef]

Rogiest, W.

W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
[CrossRef]

W. Rogiest, K. Laevens, D. Fiems, H. Bruneel, “A performance model for an asynchronous optical buffer,” Perform. Eval., vol. 62, no. 1–4, pp. 313–330, 2005.
[CrossRef]

Sabido, D. J. M.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Scheinhardt, W.

M. Mandjes, D. Mitra, W. Scheinhardt, “Models of network access using feedback fluid queues,” Queueing Syst., vol. 44, no. 4, pp. 2989–3002, 2003.
[CrossRef]

I. Adan, E. van Doorn, J. Resing, W. Scheinhardt, “Analysis of a single-server queue interacting with a fluid reservoir,” Queueing Syst., vol. 29, pp. 313–336, 1998.
[CrossRef]

W. Scheinhardt, “Markov modulated and feedback fluid queues,” Ph.D. dissertation, University of Twente, Enschede, The Netherlands, 1998.

Schilling, M.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Sondhi, M. M.

D. Anick, D. Mitra, M. M. Sondhi, “Stochastic theory of a data handling system with multiple sources,” Bell Syst. Tech. J., vol. 61, pp. 1871–1894, 1982.
[CrossRef]

Suh, C.-J.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Sung, D.

S. Kang, D. Sung, “Two-state MMPP modeling of ATM superposed traffic streams based on the characterization of correlated interarrival times,” in IEEE Global Telecommunications Conf., 1995. GLOBECOM ’95, vol. 2, Nov. 1995, pp. 1422–1426.

Talneau, A.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Tucker, R.

R. Tucker, “Accurate method for analysis of a packet speech multiplexer with limited delay,” IEEE Trans. Commun., vol. 36, no. 4, pp. 479–483, 1988.
[CrossRef]

van Doorn, E.

I. Adan, E. van Doorn, J. Resing, W. Scheinhardt, “Analysis of a single-server queue interacting with a fluid reservoir,” Queueing Syst., vol. 29, pp. 313–336, 1998.
[CrossRef]

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins U. Press, 1996.

Waldman, H.

R. C. Almeida, J. Pelegrini, H. Waldman, “A generic-traffic optical buffer modeling for asynchronous optical switching networks,” IEEE Commun. Lett., vol. 9, no. 2, pp. 175–177, Feb. 2005.
[CrossRef]

Wong, E.

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

Xiao, S.

Yoo, M.

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

Zhang, T.

T. Zhang, K. Lu, J. Jue, “Shared fiber delay line buffers in asynchronous optical packet switches,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 118–127, 2006.
[CrossRef]

Zucchelli, L.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

Bell Syst. Tech. J. (1)

D. Anick, D. Mitra, M. M. Sondhi, “Stochastic theory of a data handling system with multiple sources,” Bell Syst. Tech. J., vol. 61, pp. 1871–1894, 1982.
[CrossRef]

IEEE Commun. Lett. (2)

F. Callegati, “Optical buffers for variable length packets,” IEEE Commun. Lett., vol. 4, no. 9, pp. 292–294, Sept. 2000.
[CrossRef]

R. C. Almeida, J. Pelegrini, H. Waldman, “A generic-traffic optical buffer modeling for asynchronous optical switching networks,” IEEE Commun. Lett., vol. 9, no. 2, pp. 175–177, Feb. 2005.
[CrossRef]

IEEE J. Sel. Areas Commun. (4)

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. Gravey, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krahenbuhl, C. Raffaelli, M. Schilling, A. Talneau, L. Zucchelli, “Transparent optical packet switching: network architecture and demonstrators in the KEOPS project,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1245–1259, 1998.
[CrossRef]

R. Barry, P. Humblet, “Models of blocking probability in all-optical networks with and without wavelength changers,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 858–867, June 1996.
[CrossRef]

I. Chlamtac, A. Fumagalli, L. Kazovsky, P. Melman, W. Nelson, P. Poggiolini, M. Cerisola, A. Choudhury, T. Fong, R. Hofmeister, C.-L. Lu, A. Mekkittikul, D. J. M. Sabido, C.-J. Suh, E. Wong, “Cord: contention resolution by delay lines,” IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp. 1014–1029, June 1996.
[CrossRef]

T. Zhang, K. Lu, J. Jue, “Shared fiber delay line buffers in asynchronous optical packet switches,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 118–127, 2006.
[CrossRef]

IEEE Trans. Commun. (1)

R. Tucker, “Accurate method for analysis of a packet speech multiplexer with limited delay,” IEEE Trans. Commun., vol. 36, no. 4, pp. 479–483, 1988.
[CrossRef]

IEEE/ACM Trans. Netw. (1)

V. Paxson, S. Floyd, “Wide-area traffic: the failure of Poisson modeling,” IEEE/ACM Trans. Netw., vol. 3, pp. 226–244, 1995.
[CrossRef]

J. High Speed Networks (JHSN) (1)

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

J. Lightwave Technol. (2)

Perform. Eval. (2)

W. Rogiest, K. Laevens, D. Fiems, H. Bruneel, “A performance model for an asynchronous optical buffer,” Perform. Eval., vol. 62, no. 1–4, pp. 313–330, 2005.
[CrossRef]

W. Rogiest, J. Lambert, D. Fiems, B. V. Houdt, H. Bruneel, C. Blondia, “A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution,” Perform. Eval., vol. 66, no. 7, pp. 343–355, 2009.
[CrossRef]

Queueing Syst. (2)

M. Mandjes, D. Mitra, W. Scheinhardt, “Models of network access using feedback fluid queues,” Queueing Syst., vol. 44, no. 4, pp. 2989–3002, 2003.
[CrossRef]

I. Adan, E. van Doorn, J. Resing, W. Scheinhardt, “Analysis of a single-server queue interacting with a fluid reservoir,” Queueing Syst., vol. 29, pp. 313–336, 1998.
[CrossRef]

Scand. J. Stat. (1)

S. Asmussen, O. Nerman, M. Olsson, “Fitting phase-type distributions via the EM algorithm,” Scand. J. Stat., vol. 23, pp. 419–441, 1996.

Stoch. Models (2)

D. M. Lucantoni, “New results for the single server queue with a batch Markovian arrival process,” Stoch. Models, vol. 7, pp. 1–46, 1991.
[CrossRef]

H. E. Kankaya, N. Akar, “Solving multi-regime feedback fluid queues,” Stoch. Models, vol. 24, no. 3, pp. 425–450, July 2008.
[CrossRef]

Other (7)

M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications. New York: Marcel Dekker, 1989.

L. Muscariello, M. Mellia, M. Meo, R. L. Cigno, M. A. Marsan, “An MMPP-based hierarchical model of Internet traffic,” in 2004 IEEE Int. Conf. Communications, vol. 4, June 2004, pp. 2143–2147.
[CrossRef]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models. Baltimore, MD: Johns Hopkins U. Press, 1981.

W. Scheinhardt, “Markov modulated and feedback fluid queues,” Ph.D. dissertation, University of Twente, Enschede, The Netherlands, 1998.

B. R. Haverkort, Performance of Computer and Communication Systems: A Model-based Approach. Wiley, 1998.
[CrossRef]

S. Kang, D. Sung, “Two-state MMPP modeling of ATM superposed traffic streams based on the characterization of correlated interarrival times,” in IEEE Global Telecommunications Conf., 1995. GLOBECOM ’95, vol. 2, Nov. 1995, pp. 1422–1426.

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins U. Press, 1996.

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Figures (9)

Fig. 1
Fig. 1

Evolution of the channel horizon H ( t ) for a single-wavelength degenerate optical buffer with D = K = 2 .

Fig. 2
Fig. 2

The multiregime feedback fluid queue.

Fig. 3
Fig. 3

Evolution of the transformed process H T ( t ) for a single-wavelength degenerate optical buffer with D = K = 2 .

Fig. 4
Fig. 4

Packet blocking probability P b as a function of the granularity parameter D for varying values of the buffer size K.

Fig. 5
Fig. 5

PDF, g ̂ ( y ) , of the unfinished work that an arriving packet finds for Example 2 for varying values of a.

Fig. 6
Fig. 6

Blocking probability P b for K = 20 as a function of the granularity parameter D for autocorrelation with different values of packet length SCV (a) ρ = 0.8 , (b) ρ = 0.6 , (c) ρ = 0.4 .

Fig. 7
Fig. 7

The plot of (a) λ ( D , K , P b ) ( 1 P b ) as a function of D for various K for a desired blocking probability of P b = 10 4 . (b) The quantity D * K as a function of the buffer size K for two different values of P b and packet length SCV.

Fig. 8
Fig. 8

The plot of (a) λ 0 ( L , K , P b ) ( 1 P b ) as a function of K for various values of the total fiber size L when P b = 10 4 and SVC = 1 and (b) ( K * ) 3 2 L as a function of the total fiber size L for two different values of P b and packet length SCV.

Fig. 9
Fig. 9

Blocking probability P b against D for various values of σ when λ = 0.25 .

Tables (2)

Tables Icon

Table 1 Three-Step Procedure for Obtaining the Transformed Process H T ( t )

Tables Icon

Table 2 The Optimum Granularity Parameter D * for Various Values of K and for Two Values of λ

Equations (48)

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

D H ( t ) D ,
H ( t + ) = D H ( t ) D + B .
Q PH = [ S S 0 0 0 ] ,
d C ( t ) d t = { max ( 0 , r ̃ M ( t ) ( 0 ) ) if C ( t ) = 0 , r M ( t ) ( k ) if T ( k 1 ) < C ( t ) < T ( k ) , r ̃ M ( t ) ( k ) if C ( t ) = T ( k ) . }
f m ( k ) ( y , t ) = F m ( k ) ( y , t ) y , T ( k 1 ) < y < T ( k ) ,
F m ( k ) ( y , t ) = P ( C ( t ) y , M ( t ) = m ) , T ( k 1 ) < y < T ( k ) ,
f ( k ) ( y ) = [ f 1 ( k ) ( y ) , f 2 ( k ) ( y ) , , f M ( k ) ( y ) ] .
c ( k ) ( t ) = [ c 1 ( k ) ( t ) , c 2 ( k ) ( t ) , , c M ( k ) ( t ) ] ,
c m ( k ) ( t ) = P ( C ( t ) = T ( k ) , M ( t ) = m ) ,
F m ( y ) = F m ( k ) ( y ) , 1 m M , when T ( k 1 ) < y < T ( k ) ,
F ( y ) = [ F 1 ( y ) , F 2 ( y ) , , F M ( y ) ] .
R ( K + 1 ) = diag ( I d , I d , I s d ) ,
R ( k ) = diag ( I d , I d , I s d ) , 1 k K ,
R ̃ ( 0 ) = diag ( 0 d , I d , I s d ) ,
R ̃ ( k ) = diag ( I d , 0 d , I s d ) , 1 k K ,
Q ( k ) = [ D 0 D 1 0 0 0 0 S ̃ 0 0 S ̃ ] , k K ,
Q ( K + 1 ) = [ D ̃ 0 0 0 0 0 S ̃ 0 0 S ̃ ] ,
Q ̃ ( k ) = [ D 0 D 1 0 0 I d V ̃ S ̃ 0 0 S ̃ ] , k > 0 ,
Q ̃ ( 0 ) = [ D 0 0 D ̃ 1 0 0 0 S ̃ 0 0 S ̃ ] ,
G j ( y ) = F j ( y ) j = 1 d F j ( ) ,
G ̃ j ( y ) = G j ( ) G j ( y ) , 1 j d .
P b = G ̃ ( K D ) D 1 e π D 1 e .
E [ Delay ] = k = 1 K k D ( G ̂ ( k D ) G ̂ ( ( k 1 ) D ) ) 1 P b ,
G ̂ ( y ) = G ( y ) D 1 e π D 1 e
S = [ 20 10 10 10 ] , v = [ 0.5 0.5 ] ,
D 0 = 1 a [ 5 1 1 1 5 1 1 1 5 ] , D 1 = 1 a [ 1 2 0 1 1 1 1 1 1 ] ,
D * K 1 2
K * L 2 3
f ( k + 1 ) ( T ( k ) + ) R ( k + 1 ) f ( k ) ( T ( k ) ) R ( k ) = c ( k ) Q ̃ ( k ) , 1 k K ,
f m ( k + 1 ) ( T ( k ) + ) = 0 , m B , 1 k K ,
c m ( k ) = 0 , m A C , 1 k K ,
f ( 1 ) ( 0 + ) R ( 1 ) = c ( 0 ) Q ̃ ( 0 ) ,
c m ( 0 ) = 0 , m C .
A ( k ) = A = [ D 0 D 1 0 0 0 0 S ̃ 0 0 S ̃ ] , 1 k K
= A ̃ = [ D ̃ 0 0 0 0 0 S ̃ 0 0 S ̃ ] , k = K + 1 .
S 1 = [ S ̃ 0 0 ] ,
S 2 = [ D 0 D 1 0 0 ] ,
S ̃ 2 = [ D ̃ 0 0 0 ] ,
S ̃ X 1 + X 1 S 2 + S 1 = 0 ,
S ̃ X ̃ 1 + X ̃ 1 S ̃ 2 + S 1 = 0 .
Y = [ X 11 X 12 I I X 2 0 0 I 0 ] ,
Y ̃ = [ X ̃ 11 X ̃ 12 I I 0 0 0 I 0 ]
Y A Y 1 = diag ( S ̃ , D 0 , 0 d ) ,
Y ̃ A ̃ Y ̃ 1 = diag ( S ̃ , D ̃ , 0 d ) .
f ( k ) ( y ) = a ( k ) e S ̃ ( y T ( k 1 ) ) L + a + ( k ) e D 0 ( T ( k ) y ) L + + a 0 ( k ) L 0 , 1 k K ,
f ( K + 1 ) ( y ) = a ( K + 1 ) e S ̃ ( y T ( K ) ) L ̃ ,
m = 1 M ( k = 1 K + 1 T ( k 1 ) + T ( k ) f m ( k ) ( x ) d x + k = 0 K c m ( k ) ) = 1 .
1 = k = 0 K c ( k ) e + k = 1 K a 0 ( k ) L 0 e + k = 1 K a ( k ) ( e S ̃ Δ ( K ) I ) S ̃ 1 L e a ( K + 1 ) S ̃ 1 L ̃ e + k = 1 K a + ( k ) ( e D 0 Δ ( K ) I ) D 0 1 L + e ,