Abstract

Management of quality of service in optical packet switches equipped with fixed input, tunable output wavelength converters is described. This kind of switch is considered with the aim to keep switch cost low by employing simple shared optical components and low-complexity space switching matrices. In particular, differentiation among classes of service in terms of loss performance for bufferless switches is addressed. The functionalities described refer to the high-speed multiservice networking context foreseeable for the future Internet. A switch control algorithm to manage service classes in a slotted multiplexing context is described. An analytical model to evaluate the loss probability of each class is developed and validated against a simulation and is used to prove the effectiveness of the quality-of-service approach. Numerical results show that the proposed scheme achieves basic class isolation as needed in core network optical packet switches.

© 2009 Optical Society of America

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References

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  1. M. J. O’Mahony, C. Politi, D. Klonidis, R. Nejabati, D. Simeonidou, “Future optical network,” J. Lightwave Technol., vol. 24, no. 12, pp. 4684–4696, Dec. 2006.
    [CrossRef]
  2. X. Xiao, L. M. Ni, “Internet QoS: a big picture,” IEEE Network, vol. 13, pp. 8–18, 1999.
    [CrossRef]
  3. M. J. O’Mahony, D. Simeonidou, D. Hunter, A. Tzanakaki, “The application of optical packet switching in future communication network,” IEEE Commun. Mag., vol. 39, no. 3, pp. 128–135, March 2001.
    [CrossRef]
  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. 1245–1253, Sept. 1998.
    [CrossRef]
  5. L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
    [CrossRef]
  6. R. Braden, D. Clark, S. Shenker, “Integrated services in the internet architecture: an overview,” IETF RFC 1633, July 1994.
  7. S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, “An architecture for differentiated services,” IETF RFC 2475, Dec. 1998.
  8. H. Overby, N. Stol, M. Nord, “Evaluation of QoS differentiation mechanism in asynchronous bufferless optical packet-switched networks,” IEEE Commun. Mag., vol. 44, no. 8, pp. 52–57, Aug. 2006.
    [CrossRef]
  9. H. Overby, “Packet loss rate differentiation in slotted optical packet switched networks,” IEEE Photon. Technol. Lett., vol. 17, no. 11, pp. 2469–2471, Nov. 2005.
    [CrossRef]
  10. V. Eramo, M. Listanti, G. Pacifici, “A comparison study on the number of wavelength converters needed in synchronous and asynchronous all-optical switching architectures,” J. Lightwave Technol., vol. 21, no. 2, pp. 340–355, Feb. 2003.
    [CrossRef]
  11. V. Eramo, M. Listanti, R. Tiberio, “Performance evaluation of QoS-aware optical packet switches,” in IEEE Int. Conf. on Communications, 2008. ICC '08, Beijing, China, May 19–23, 2008, pp. 5315–5320.
  12. C. Raffaelli, M. Savi, A. Stavdas, “Sharing wavelength converters in multistage optical packet switches,” presented at 2006 Workshop on High Performance Switching and Routing, Poznan, Poland, June 7–9, 2006.
  13. V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Multifiber shared-per-wavelength all-optical switching: architectures, control, and performance,” J. Lightwave Technol., vol. 26, no. 5, pp. 537–551, March 2008.
    [CrossRef]
  14. V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Packet loss analysis of shared-per wavelength multi-fiber all-optical switch with parallel scheduling,” Comput. Netw., vol. 53, no. 2, pp. 202–216, Feb. 2009.
    [CrossRef]
  15. V. Eramo, A. Germoni, A. Cianfrani, C. Raffaelli, M. Savi, “Evaluation of QoS differentiation mechanism in shared-per-wavelength optical packet switches,” presented at 13th Conf. on Optical Network Design and Modeling, ONDM 2009, Braunschweig, Germany, Feb. 18–20, 2009.
  16. H. Li, L.-J. Thng, “Performance analysis of limited number of wavelength converters by share per node in optical switching network,” Comput. Netw., vol. 51, no. 3, pp. 671–682, Feb. 2007.
    [CrossRef]
  17. W. Feller, An Introduction to Probability Theory and its Applications, New York, NY: Wiley, 1968, vol. 1.

2009 (1)

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Packet loss analysis of shared-per wavelength multi-fiber all-optical switch with parallel scheduling,” Comput. Netw., vol. 53, no. 2, pp. 202–216, Feb. 2009.
[CrossRef]

2008 (1)

2007 (1)

H. Li, L.-J. Thng, “Performance analysis of limited number of wavelength converters by share per node in optical switching network,” Comput. Netw., vol. 51, no. 3, pp. 671–682, Feb. 2007.
[CrossRef]

2006 (2)

H. Overby, N. Stol, M. Nord, “Evaluation of QoS differentiation mechanism in asynchronous bufferless optical packet-switched networks,” IEEE Commun. Mag., vol. 44, no. 8, pp. 52–57, Aug. 2006.
[CrossRef]

M. J. O’Mahony, C. Politi, D. Klonidis, R. Nejabati, D. Simeonidou, “Future optical network,” J. Lightwave Technol., vol. 24, no. 12, pp. 4684–4696, Dec. 2006.
[CrossRef]

2005 (1)

H. Overby, “Packet loss rate differentiation in slotted optical packet switched networks,” IEEE Photon. Technol. Lett., vol. 17, no. 11, pp. 2469–2471, Nov. 2005.
[CrossRef]

2003 (2)

V. Eramo, M. Listanti, G. Pacifici, “A comparison study on the number of wavelength converters needed in synchronous and asynchronous all-optical switching architectures,” J. Lightwave Technol., vol. 21, no. 2, pp. 340–355, Feb. 2003.
[CrossRef]

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

2001 (1)

M. J. O’Mahony, D. Simeonidou, D. Hunter, A. Tzanakaki, “The application of optical packet switching in future communication network,” IEEE Commun. Mag., vol. 39, no. 3, pp. 128–135, March 2001.
[CrossRef]

1999 (1)

X. Xiao, L. M. Ni, “Internet QoS: a big picture,” IEEE Network, vol. 13, pp. 8–18, 1999.
[CrossRef]

1998 (1)

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. 1245–1253, Sept. 1998.
[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. 1245–1253, Sept. 1998.
[CrossRef]

Berde, B.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Black, D.

S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, “An architecture for differentiated services,” IETF RFC 2475, Dec. 1998.

Blake, S.

S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, “An architecture for differentiated services,” IETF RFC 2475, Dec. 1998.

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. 1245–1253, Sept. 1998.
[CrossRef]

Braden, R.

R. Braden, D. Clark, S. Shenker, “Integrated services in the internet architecture: an overview,” IETF RFC 1633, July 1994.

Callegati, F.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[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. 1245–1253, Sept. 1998.
[CrossRef]

Carlson, M.

S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, “An architecture for differentiated services,” IETF RFC 2475, Dec. 1998.

Cerroni, W.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Chiaroni, D.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[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. 1245–1253, Sept. 1998.
[CrossRef]

Cianfrani, A.

V. Eramo, A. Germoni, A. Cianfrani, C. Raffaelli, M. Savi, “Evaluation of QoS differentiation mechanism in shared-per-wavelength optical packet switches,” presented at 13th Conf. on Optical Network Design and Modeling, ONDM 2009, Braunschweig, Germany, Feb. 18–20, 2009.

Clark, D.

R. Braden, D. Clark, S. Shenker, “Integrated services in the internet architecture: an overview,” IETF RFC 1633, July 1994.

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. 1245–1253, Sept. 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. 1245–1253, Sept. 1998.
[CrossRef]

Davies, E.

S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, “An architecture for differentiated services,” IETF RFC 2475, Dec. 1998.

Dembeck, L.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Develder, C.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Dittmann, L.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Eilenberger, G.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Eramo, V.

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Packet loss analysis of shared-per wavelength multi-fiber all-optical switch with parallel scheduling,” Comput. Netw., vol. 53, no. 2, pp. 202–216, Feb. 2009.
[CrossRef]

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Multifiber shared-per-wavelength all-optical switching: architectures, control, and performance,” J. Lightwave Technol., vol. 26, no. 5, pp. 537–551, March 2008.
[CrossRef]

V. Eramo, M. Listanti, G. Pacifici, “A comparison study on the number of wavelength converters needed in synchronous and asynchronous all-optical switching architectures,” J. Lightwave Technol., vol. 21, no. 2, pp. 340–355, Feb. 2003.
[CrossRef]

V. Eramo, M. Listanti, R. Tiberio, “Performance evaluation of QoS-aware optical packet switches,” in IEEE Int. Conf. on Communications, 2008. ICC '08, Beijing, China, May 19–23, 2008, pp. 5315–5320.

V. Eramo, A. Germoni, A. Cianfrani, C. Raffaelli, M. Savi, “Evaluation of QoS differentiation mechanism in shared-per-wavelength optical packet switches,” presented at 13th Conf. on Optical Network Design and Modeling, ONDM 2009, Braunschweig, Germany, Feb. 18–20, 2009.

Feller, W.

W. Feller, An Introduction to Probability Theory and its Applications, New York, NY: Wiley, 1968, vol. 1.

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. 1245–1253, Sept. 1998.
[CrossRef]

Germoni, A.

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Packet loss analysis of shared-per wavelength multi-fiber all-optical switch with parallel scheduling,” Comput. Netw., vol. 53, no. 2, pp. 202–216, Feb. 2009.
[CrossRef]

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Multifiber shared-per-wavelength all-optical switching: architectures, control, and performance,” J. Lightwave Technol., vol. 26, no. 5, pp. 537–551, March 2008.
[CrossRef]

V. Eramo, A. Germoni, A. Cianfrani, C. Raffaelli, M. Savi, “Evaluation of QoS differentiation mechanism in shared-per-wavelength optical packet switches,” presented at 13th Conf. on Optical Network Design and Modeling, ONDM 2009, Braunschweig, Germany, Feb. 18–20, 2009.

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. 1245–1253, Sept. 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. 1245–1253, Sept. 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. 1245–1253, Sept. 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. 1245–1253, Sept. 1998.
[CrossRef]

Hunter, D.

M. J. O’Mahony, D. Simeonidou, D. Hunter, A. Tzanakaki, “The application of optical packet switching in future communication network,” IEEE Commun. Mag., vol. 39, no. 3, pp. 128–135, March 2001.
[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. 1245–1253, Sept. 1998.
[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. 1245–1253, Sept. 1998.
[CrossRef]

Klonidis, D.

Koerber, W.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[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. 1245–1253, Sept. 1998.
[CrossRef]

Le Sauze, N.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Leligou, N.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Li, H.

H. Li, L.-J. Thng, “Performance analysis of limited number of wavelength converters by share per node in optical switching network,” Comput. Netw., vol. 51, no. 3, pp. 671–682, Feb. 2007.
[CrossRef]

Listanti, M.

V. Eramo, M. Listanti, G. Pacifici, “A comparison study on the number of wavelength converters needed in synchronous and asynchronous all-optical switching architectures,” J. Lightwave Technol., vol. 21, no. 2, pp. 340–355, Feb. 2003.
[CrossRef]

V. Eramo, M. Listanti, R. Tiberio, “Performance evaluation of QoS-aware optical packet switches,” in IEEE Int. Conf. on Communications, 2008. ICC '08, Beijing, China, May 19–23, 2008, pp. 5315–5320.

Mahony, M.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Mortensen, B.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Nejabati, R.

Neri, F.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Ni, L. M.

X. Xiao, L. M. Ni, “Internet QoS: a big picture,” IEEE Network, vol. 13, pp. 8–18, 1999.
[CrossRef]

Nord, M.

H. Overby, N. Stol, M. Nord, “Evaluation of QoS differentiation mechanism in asynchronous bufferless optical packet-switched networks,” IEEE Commun. Mag., vol. 44, no. 8, pp. 52–57, Aug. 2006.
[CrossRef]

O’Mahony, M. J.

M. J. O’Mahony, C. Politi, D. Klonidis, R. Nejabati, D. Simeonidou, “Future optical network,” J. Lightwave Technol., vol. 24, no. 12, pp. 4684–4696, Dec. 2006.
[CrossRef]

M. J. O’Mahony, D. Simeonidou, D. Hunter, A. Tzanakaki, “The application of optical packet switching in future communication network,” IEEE Commun. Mag., vol. 39, no. 3, pp. 128–135, March 2001.
[CrossRef]

Overby, H.

H. Overby, N. Stol, M. Nord, “Evaluation of QoS differentiation mechanism in asynchronous bufferless optical packet-switched networks,” IEEE Commun. Mag., vol. 44, no. 8, pp. 52–57, Aug. 2006.
[CrossRef]

H. Overby, “Packet loss rate differentiation in slotted optical packet switched networks,” IEEE Photon. Technol. Lett., vol. 17, no. 11, pp. 2469–2471, Nov. 2005.
[CrossRef]

Pacifici, G.

Pickavet, M.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Politi, C.

Rafel, A.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Raffaelli, C.

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Packet loss analysis of shared-per wavelength multi-fiber all-optical switch with parallel scheduling,” Comput. Netw., vol. 53, no. 2, pp. 202–216, Feb. 2009.
[CrossRef]

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Multifiber shared-per-wavelength all-optical switching: architectures, control, and performance,” J. Lightwave Technol., vol. 26, no. 5, pp. 537–551, March 2008.
[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. 1245–1253, Sept. 1998.
[CrossRef]

C. Raffaelli, M. Savi, A. Stavdas, “Sharing wavelength converters in multistage optical packet switches,” presented at 2006 Workshop on High Performance Switching and Routing, Poznan, Poland, June 7–9, 2006.

V. Eramo, A. Germoni, A. Cianfrani, C. Raffaelli, M. Savi, “Evaluation of QoS differentiation mechanism in shared-per-wavelength optical packet switches,” presented at 13th Conf. on Optical Network Design and Modeling, ONDM 2009, Braunschweig, Germany, Feb. 18–20, 2009.

Renaud, M.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[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. 1245–1253, Sept. 1998.
[CrossRef]

Savi, M.

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Packet loss analysis of shared-per wavelength multi-fiber all-optical switch with parallel scheduling,” Comput. Netw., vol. 53, no. 2, pp. 202–216, Feb. 2009.
[CrossRef]

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Multifiber shared-per-wavelength all-optical switching: architectures, control, and performance,” J. Lightwave Technol., vol. 26, no. 5, pp. 537–551, March 2008.
[CrossRef]

C. Raffaelli, M. Savi, A. Stavdas, “Sharing wavelength converters in multistage optical packet switches,” presented at 2006 Workshop on High Performance Switching and Routing, Poznan, Poland, June 7–9, 2006.

V. Eramo, A. Germoni, A. Cianfrani, C. Raffaelli, M. Savi, “Evaluation of QoS differentiation mechanism in shared-per-wavelength optical packet switches,” presented at 13th Conf. on Optical Network Design and Modeling, ONDM 2009, Braunschweig, Germany, Feb. 18–20, 2009.

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. 1245–1253, Sept. 1998.
[CrossRef]

Shenker, S.

R. Braden, D. Clark, S. Shenker, “Integrated services in the internet architecture: an overview,” IETF RFC 1633, July 1994.

Simeonidou, D.

M. J. O’Mahony, C. Politi, D. Klonidis, R. Nejabati, D. Simeonidou, “Future optical network,” J. Lightwave Technol., vol. 24, no. 12, pp. 4684–4696, Dec. 2006.
[CrossRef]

M. J. O’Mahony, D. Simeonidou, D. Hunter, A. Tzanakaki, “The application of optical packet switching in future communication network,” IEEE Commun. Mag., vol. 39, no. 3, pp. 128–135, March 2001.
[CrossRef]

Sole-Pareta, J.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

Stavdas, A.

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

C. Raffaelli, M. Savi, A. Stavdas, “Sharing wavelength converters in multistage optical packet switches,” presented at 2006 Workshop on High Performance Switching and Routing, Poznan, Poland, June 7–9, 2006.

Stol, N.

H. Overby, N. Stol, M. Nord, “Evaluation of QoS differentiation mechanism in asynchronous bufferless optical packet-switched networks,” IEEE Commun. Mag., vol. 44, no. 8, pp. 52–57, Aug. 2006.
[CrossRef]

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. 1245–1253, Sept. 1998.
[CrossRef]

Thng, L.-J.

H. Li, L.-J. Thng, “Performance analysis of limited number of wavelength converters by share per node in optical switching network,” Comput. Netw., vol. 51, no. 3, pp. 671–682, Feb. 2007.
[CrossRef]

Tiberio, R.

V. Eramo, M. Listanti, R. Tiberio, “Performance evaluation of QoS-aware optical packet switches,” in IEEE Int. Conf. on Communications, 2008. ICC '08, Beijing, China, May 19–23, 2008, pp. 5315–5320.

Tzanakaki, A.

M. J. O’Mahony, D. Simeonidou, D. Hunter, A. Tzanakaki, “The application of optical packet switching in future communication network,” IEEE Commun. Mag., vol. 39, no. 3, pp. 128–135, March 2001.
[CrossRef]

Wang, Z.

S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, “An architecture for differentiated services,” IETF RFC 2475, Dec. 1998.

Weiss, W.

S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, “An architecture for differentiated services,” IETF RFC 2475, Dec. 1998.

Xiao, X.

X. Xiao, L. M. Ni, “Internet QoS: a big picture,” IEEE Network, vol. 13, pp. 8–18, 1999.
[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. 1245–1253, Sept. 1998.
[CrossRef]

Comput. Netw. (2)

V. Eramo, A. Germoni, C. Raffaelli, M. Savi, “Packet loss analysis of shared-per wavelength multi-fiber all-optical switch with parallel scheduling,” Comput. Netw., vol. 53, no. 2, pp. 202–216, Feb. 2009.
[CrossRef]

H. Li, L.-J. Thng, “Performance analysis of limited number of wavelength converters by share per node in optical switching network,” Comput. Netw., vol. 51, no. 3, pp. 671–682, Feb. 2007.
[CrossRef]

IEEE Commun. Mag. (2)

M. J. O’Mahony, D. Simeonidou, D. Hunter, A. Tzanakaki, “The application of optical packet switching in future communication network,” IEEE Commun. Mag., vol. 39, no. 3, pp. 128–135, March 2001.
[CrossRef]

H. Overby, N. Stol, M. Nord, “Evaluation of QoS differentiation mechanism in asynchronous bufferless optical packet-switched networks,” IEEE Commun. Mag., vol. 44, no. 8, pp. 52–57, Aug. 2006.
[CrossRef]

IEEE J. Sel. Areas Commun. (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. 1245–1253, Sept. 1998.
[CrossRef]

L. Dittmann, C. Develder, D. Chiaroni, F. Neri, F. Callegati, W. Koerber, A. Stavdas, M. Renaud, A. Rafel, J. Sole-Pareta, W. Cerroni, N. Leligou, L. Dembeck, B. Mortensen, M. Pickavet, N. Le Sauze, M. Mahony, B. Berde, G. Eilenberger, “The European IST project DAVID: a viable approach towards optical packet switching,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1026–1040, Sept. 2003.
[CrossRef]

IEEE Network (1)

X. Xiao, L. M. Ni, “Internet QoS: a big picture,” IEEE Network, vol. 13, pp. 8–18, 1999.
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. Overby, “Packet loss rate differentiation in slotted optical packet switched networks,” IEEE Photon. Technol. Lett., vol. 17, no. 11, pp. 2469–2471, Nov. 2005.
[CrossRef]

J. Lightwave Technol. (3)

Other (6)

W. Feller, An Introduction to Probability Theory and its Applications, New York, NY: Wiley, 1968, vol. 1.

V. Eramo, A. Germoni, A. Cianfrani, C. Raffaelli, M. Savi, “Evaluation of QoS differentiation mechanism in shared-per-wavelength optical packet switches,” presented at 13th Conf. on Optical Network Design and Modeling, ONDM 2009, Braunschweig, Germany, Feb. 18–20, 2009.

V. Eramo, M. Listanti, R. Tiberio, “Performance evaluation of QoS-aware optical packet switches,” in IEEE Int. Conf. on Communications, 2008. ICC '08, Beijing, China, May 19–23, 2008, pp. 5315–5320.

C. Raffaelli, M. Savi, A. Stavdas, “Sharing wavelength converters in multistage optical packet switches,” presented at 2006 Workshop on High Performance Switching and Routing, Poznan, Poland, June 7–9, 2006.

R. Braden, D. Clark, S. Shenker, “Integrated services in the internet architecture: an overview,” IETF RFC 1633, July 1994.

S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, “An architecture for differentiated services,” IETF RFC 2475, Dec. 1998.

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Figures (9)

Fig. 1
Fig. 1

SPW architecture with N input–output fibers, each carrying M wavelengths, and a pool of r w converters dedicated to each wavelength switching block.

Fig. 2
Fig. 2

General flow chart of the priority-based CA for SPW switch.

Fig. 3
Fig. 3

Flow chart of DOWCA phase for OF i.

Fig. 4
Fig. 4

Flow chart of the FTA phase for the wavelength λ k .

Fig. 5
Fig. 5

PLP of the SPW switch versus the number of WCs used for CoS i ( i = 0 , 1 , 2 ) . Both analytical and simulation results are reported. The switch parameters are N = 8 IF–OF and M = 32 wavelengths per fiber, and the total load offered to each IWC is p = 0 , 4 . Q = 3 CoSs are considered.

Fig. 6
Fig. 6

PLP of the SPW switch versus the number of WCs used for CoS i ( i = 0 , 1 , 2 ) . Both analytical and simulation results are reported. The switch parameters are N = 8 IF–OF and M = 32 wavelengths per fiber, and the total load offered to each IWC is p = 0 , 8 . Q = 3 CoSs are considered.

Fig. 7
Fig. 7

PLP of the SPW switch versus the number of FI-TOWCs used for CoS i ( i = 0 , 1 , 2 ) . Both analytical and simulation data are reported. The switch parameters are N = 32 IF–OF and M = 8 wavelengths per fiber, and the total load offered to each IWC is p = 0 , 4 . Q = 3 CoSs are considered.

Fig. 8
Fig. 8

PLP of the SPW switch versus the number of used FI-TOWCs for CoS i ( i = 0 , 1 , 2 ) . Both analytical and simulation data are reported. The switch parameters are N = 32 IF–OF and M = 8 wavelengths per fiber, and the total load offered to each IWC is p = 0 , 8 . Q = 3 CoSs are considered.

Fig. 9
Fig. 9

PLP of the SPW switch versus the number of WCs used. The switch parameters are N = 32 , M = 16 , 32 . The traffic offered to each IWC is p = 0 , 8 . Q = 3 CoSs are considered.

Tables (1)

Tables Icon

Table 1 P l , i for r w Equal to Threshold Values of the Number of FI-TOWCs for the CoS i ( i = 0 , 1 , 2 ) Packets a

Equations (16)

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

P l , i = E [ N p , i w ] + E [ N p , i c ] E [ N o , i ] .
E [ N p , i w , OF ] = { E [ N p , i w , OF ] i = 0 E [ N p , i w , OF ] E [ N p , i 1 w , OF ] i = 1 , , Q 1 } .
E [ N p , i w , OF ] = h = 0 M r = 0 M h s = 0 N ( M h r ) E [ N p , i w , OF α = h , β i = r , γ i = s ] p α , β i , γ i ( h , r , s ) .
E [ N p , i w , OF α = h , β i = r , γ i = s ] = max ( s ( M h r ) h , 0 ) = max ( s M + r , 0 ) .
E [ N p , i c , λ ] = { E [ N p , i c , λ ] i = 0 E [ N p , i c , λ ] E [ N p , i 1 c , λ ] i = 1 , , Q 1 } .
E [ N p , i c , λ ] h = r w + 1 M ( N 1 ) ( h r w ) p W i λ ( h ) ,
ϑ = ( 1 p N ) N .
ϕ i = h = 1 N ( N h ) ( q > i p N ) h ( 1 p N ) N h .
p α , β i , γ i ( h , r , s ) = p γ i α , β i ( s ) p α , β i ( h , r ) ,
p α , β i ( h , r ) = ( M h ) ( M h r ) ϑ h ϕ i r ( 1 ( ϑ + ϕ i ) ) M h r , h = 0 , , M , r = 0 , , M h .
p γ i α , β i ( s ) = Pr { γ i = s α = h , β i = r } = Pr { j = 1 M h r X σ j X σ 1 1 , X σ 2 1 , , X σ M r h 1 } .
p γ i α , β i ( s ) = { p X σ 1 X σ 1 1 ( s ) p X σ M h r X σ M h r 1 ( s ) s = M h r , , N ( M h r ) 0 s = 0 , , M h r } ,
p X σ j X σ j 1 ( s ) = Pr { X σ j = s , X σ j 1 } Pr { X σ j 1 } = { ( N s ) ( q i p N ) s ( 1 q i p N ) N s 1 ( 1 q i p N ) N s = 1 , , N 0 s = 0 } .
W i λ = R i λ ( N N i λ ) ,
p W i λ ( h ) = j = 0 N ( N j ) q i j ( 1 q i j ) N j Pr ( N i λ = h + N j R i λ = j ) .
Pr ( N i λ = k R i λ = j ) = ( N k ) ν = 0 N k ( 1 ) ν ( N k ν ) ( 1 k + ν N ) j .