Abstract

Optical buffering is a major challenge in realizing all-optical packet switching. In this paper we propose a new buffer called a multiple-input single-output FIFO (MISO-FIFO) optical buffer that supports several functions normally associated with electronic RAM. Our structure reduces the physical size of a buffer by up to an order of magnitude or more by allowing reuse of its basic optical delay line (ODL) elements. Moreover, by using controllable fractional delay lines (CFDLs) as the basic building block we are able to reduce the size and frequency of voids in the output of the buffer. We develop a Markov Chain (MC) model for the performance of our new buffering scheme, and demonstrate the advantages of our structure over buffer structures that use ODLs in terms of throughput and link utilization.

© 2008 Optical Society of America

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  6. R. S. Tucker et al., "Slow-light optical buffers: capabilities and fundamental limitations," IEEE/OSA J. Lightwave Technol. 23-12, 4046-4066 (2005).
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  7. R. S. Tucker, "The role of optics and electronics in high capacity routers," IEEE/OSA J. Lightwave Technol. 24-12, 4655-4673 (2006).
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    [CrossRef]
  20. T. Zhang, K. Lu, and J. R. Jue, "Shared fiber delay line buffers in asynchronous optical packet switches," IEEE J. Sel. Areas Commun. 24-4, 118-127 (2006).
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2006 (2)

R. S. Tucker, "The role of optics and electronics in high capacity routers," IEEE/OSA J. Lightwave Technol. 24-12, 4655-4673 (2006).
[CrossRef]

T. Zhang, K. Lu, and J. R. Jue, "Shared fiber delay line buffers in asynchronous optical packet switches," IEEE J. Sel. Areas Commun. 24-4, 118-127 (2006).
[CrossRef]

2005 (3)

R. S. Tucker et al., "Slow-light optical buffers: capabilities and fundamental limitations," IEEE/OSA J. Lightwave Technol. 23-12, 4046-4066 (2005).
[CrossRef]

N. McKeown et al., "Part III: routers with very small buffers," ACM SIGCOMM Computer Communication Review 35-2, 73-89 (2005).
[CrossRef]

R. C. Almeida, J. U. Pelegrini, and H. Waldman, "A generic-traffic optical buffer modeling for asynchronous optical switching networks," IEEE Commun. Lett. 9-2, 175 - 177 (2005).
[CrossRef]

2004 (1)

Y. K. Yeo et al., "A dynamically reconfigurable folded-path time delay buffer for optical packet switching," IEEE Photon. Technol. Lett. 16-11, 2559-2561 (2004).
[CrossRef]

2003 (1)

X. Zhu and J. M. Khan "Queuing models of optical delay lines in synchronous and asynchronous optical packet-switching networks,' Optical Engin. 42-6, 1741-1748 (2003).
[CrossRef]

2001 (2)

M. C. Chia, D. K. Hunter et al., "Packet loss and delay performance of feedback and feed-forward arrayed-waveguide gratings-based optical packet switches with WDM inputs-outputs," IEEE/OSA J. Lightwave Technol. 19-9, 1241 - 1254 (2001).
[CrossRef]

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, "The application of optical packet switching in future communication networks," IEEE Commun. Mag. 3, 128-135 (2001).
[CrossRef]

2000 (5)

S. Yoo and B. Mukherjee, "Advances in photonic packet switching an overview," IEEE Commun. Mag. 2, 84-94 (2000).

D. K. Hunter and I. Andonovic, "Approaches to optical internet packet switching," IEEE Commun. Mag. 9, 116-122 (2000).
[CrossRef]

L. Tancevski, S. Yegnanarayanan, G. Castanon, L. Tamil, F. Masetti, and T. McDermott, "Optical routing of asynchronous, variable length packets," IEEE J. Sel. Areas Commun. 18-10, 2084-2093 (2000).

S. H. Chin, A. Franzen, D. K. Hunter, and I. Andonovic, "Synchronisation schemes for optical networks," IEE Proc. Optoelectron 147-6, 423-427 (2000).
[CrossRef]

F. Callegati, "Optical buffers for variable length packets," IEEE Commun. Lett. 4-9, 292 - 294 (2000).
[CrossRef]

1999 (2)

P. D. BergstromJr., M. A. Ingram, A. J. Vernon, J. L. A. Hughes, and P. Tetali, "A Markov chain model for an Optical Shared-Memory Packet Switch," IEEE Trans. Commun. 47-10, 1593-1603 (1999).
[CrossRef]

D. K. Hunter et al., "WASPNET: a wavelength switched packet network," IEEE Commun. Mag. 3, 84-94 (1999).

1998 (2)

D. K. Hunter, M. C. Chia, and I. Andonovic, "Buffering in optical packet switches," IEEE/OSA J. Lightwave Technol. 16-12, 2081-1094 (1998).
[CrossRef]

P. Gambini et al., "Transparent optical packet switching: network architecture and demonstration in KEOPS project," IEEE J. Sel. Areas Commun. 16-17, 1245 - 1259 (1998).
[CrossRef]

1996 (1)

I. Chlamtac et al, "CORD: contention resolution by delay lines," IEEE J. Sel. Areas Commun. 14-5, 1014 - 1029 (1996).
[CrossRef]

1995 (1)

V. Paxson and S. Floyd, "Wide-Area Traffic: The Failure of Poisson Modeling," IEEE/ACM Trans. on Networking,  3-3, 226-244 (1995).
[CrossRef]

1987 (1)

M. J. Karol et al., "Input versus output queueing on a space-division packet switch," IEEE Trans. Commun. 35-12, 1347-1356 (1987).
[CrossRef]

Almeida, R. C.

R. C. Almeida, J. U. Pelegrini, and H. Waldman, "A generic-traffic optical buffer modeling for asynchronous optical switching networks," IEEE Commun. Lett. 9-2, 175 - 177 (2005).
[CrossRef]

Andonovic, I.

D. K. Hunter and I. Andonovic, "Approaches to optical internet packet switching," IEEE Commun. Mag. 9, 116-122 (2000).
[CrossRef]

S. H. Chin, A. Franzen, D. K. Hunter, and I. Andonovic, "Synchronisation schemes for optical networks," IEE Proc. Optoelectron 147-6, 423-427 (2000).
[CrossRef]

D. K. Hunter, M. C. Chia, and I. Andonovic, "Buffering in optical packet switches," IEEE/OSA J. Lightwave Technol. 16-12, 2081-1094 (1998).
[CrossRef]

Bergstrom, P. D.

P. D. BergstromJr., M. A. Ingram, A. J. Vernon, J. L. A. Hughes, and P. Tetali, "A Markov chain model for an Optical Shared-Memory Packet Switch," IEEE Trans. Commun. 47-10, 1593-1603 (1999).
[CrossRef]

Callegati, F.

F. Callegati, "Optical buffers for variable length packets," IEEE Commun. Lett. 4-9, 292 - 294 (2000).
[CrossRef]

Castanon, G.

L. Tancevski, S. Yegnanarayanan, G. Castanon, L. Tamil, F. Masetti, and T. McDermott, "Optical routing of asynchronous, variable length packets," IEEE J. Sel. Areas Commun. 18-10, 2084-2093 (2000).

Chia, M. C.

M. C. Chia, D. K. Hunter et al., "Packet loss and delay performance of feedback and feed-forward arrayed-waveguide gratings-based optical packet switches with WDM inputs-outputs," IEEE/OSA J. Lightwave Technol. 19-9, 1241 - 1254 (2001).
[CrossRef]

D. K. Hunter, M. C. Chia, and I. Andonovic, "Buffering in optical packet switches," IEEE/OSA J. Lightwave Technol. 16-12, 2081-1094 (1998).
[CrossRef]

Chin, S. H.

S. H. Chin, A. Franzen, D. K. Hunter, and I. Andonovic, "Synchronisation schemes for optical networks," IEE Proc. Optoelectron 147-6, 423-427 (2000).
[CrossRef]

Chlamtac, I.

I. Chlamtac et al, "CORD: contention resolution by delay lines," IEEE J. Sel. Areas Commun. 14-5, 1014 - 1029 (1996).
[CrossRef]

Floyd, S.

V. Paxson and S. Floyd, "Wide-Area Traffic: The Failure of Poisson Modeling," IEEE/ACM Trans. on Networking,  3-3, 226-244 (1995).
[CrossRef]

Franzen, A.

S. H. Chin, A. Franzen, D. K. Hunter, and I. Andonovic, "Synchronisation schemes for optical networks," IEE Proc. Optoelectron 147-6, 423-427 (2000).
[CrossRef]

Gambini, P.

P. Gambini et al., "Transparent optical packet switching: network architecture and demonstration in KEOPS project," IEEE J. Sel. Areas Commun. 16-17, 1245 - 1259 (1998).
[CrossRef]

Hughes, J. L. A.

P. D. BergstromJr., M. A. Ingram, A. J. Vernon, J. L. A. Hughes, and P. Tetali, "A Markov chain model for an Optical Shared-Memory Packet Switch," IEEE Trans. Commun. 47-10, 1593-1603 (1999).
[CrossRef]

Hunter, D. K.

M. C. Chia, D. K. Hunter et al., "Packet loss and delay performance of feedback and feed-forward arrayed-waveguide gratings-based optical packet switches with WDM inputs-outputs," IEEE/OSA J. Lightwave Technol. 19-9, 1241 - 1254 (2001).
[CrossRef]

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, "The application of optical packet switching in future communication networks," IEEE Commun. Mag. 3, 128-135 (2001).
[CrossRef]

S. H. Chin, A. Franzen, D. K. Hunter, and I. Andonovic, "Synchronisation schemes for optical networks," IEE Proc. Optoelectron 147-6, 423-427 (2000).
[CrossRef]

D. K. Hunter and I. Andonovic, "Approaches to optical internet packet switching," IEEE Commun. Mag. 9, 116-122 (2000).
[CrossRef]

D. K. Hunter et al., "WASPNET: a wavelength switched packet network," IEEE Commun. Mag. 3, 84-94 (1999).

D. K. Hunter, M. C. Chia, and I. Andonovic, "Buffering in optical packet switches," IEEE/OSA J. Lightwave Technol. 16-12, 2081-1094 (1998).
[CrossRef]

Ingram, M. A.

P. D. BergstromJr., M. A. Ingram, A. J. Vernon, J. L. A. Hughes, and P. Tetali, "A Markov chain model for an Optical Shared-Memory Packet Switch," IEEE Trans. Commun. 47-10, 1593-1603 (1999).
[CrossRef]

Jue, J. R.

T. Zhang, K. Lu, and J. R. Jue, "Shared fiber delay line buffers in asynchronous optical packet switches," IEEE J. Sel. Areas Commun. 24-4, 118-127 (2006).
[CrossRef]

Karol, M. J.

M. J. Karol et al., "Input versus output queueing on a space-division packet switch," IEEE Trans. Commun. 35-12, 1347-1356 (1987).
[CrossRef]

Khan, J. M.

X. Zhu and J. M. Khan "Queuing models of optical delay lines in synchronous and asynchronous optical packet-switching networks,' Optical Engin. 42-6, 1741-1748 (2003).
[CrossRef]

Lu, K.

T. Zhang, K. Lu, and J. R. Jue, "Shared fiber delay line buffers in asynchronous optical packet switches," IEEE J. Sel. Areas Commun. 24-4, 118-127 (2006).
[CrossRef]

Masetti, F.

L. Tancevski, S. Yegnanarayanan, G. Castanon, L. Tamil, F. Masetti, and T. McDermott, "Optical routing of asynchronous, variable length packets," IEEE J. Sel. Areas Commun. 18-10, 2084-2093 (2000).

McDermott, T.

L. Tancevski, S. Yegnanarayanan, G. Castanon, L. Tamil, F. Masetti, and T. McDermott, "Optical routing of asynchronous, variable length packets," IEEE J. Sel. Areas Commun. 18-10, 2084-2093 (2000).

McKeown, N.

N. McKeown et al., "Part III: routers with very small buffers," ACM SIGCOMM Computer Communication Review 35-2, 73-89 (2005).
[CrossRef]

Mukherjee, B.

S. Yoo and B. Mukherjee, "Advances in photonic packet switching an overview," IEEE Commun. Mag. 2, 84-94 (2000).

O’Mahony, M. J.

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, "The application of optical packet switching in future communication networks," IEEE Commun. Mag. 3, 128-135 (2001).
[CrossRef]

Pacheco, A.

P. Salvador, A. Pacheco, and R. Valadas, "Modeling IP traffic: joint characterization of packet arrivals and packet sizes using BMAPs," Elsevier J. Comput. Networks 44-3, 335-352 (2004).
[CrossRef]

Paxson, V.

V. Paxson and S. Floyd, "Wide-Area Traffic: The Failure of Poisson Modeling," IEEE/ACM Trans. on Networking,  3-3, 226-244 (1995).
[CrossRef]

Pelegrini, J. U.

R. C. Almeida, J. U. Pelegrini, and H. Waldman, "A generic-traffic optical buffer modeling for asynchronous optical switching networks," IEEE Commun. Lett. 9-2, 175 - 177 (2005).
[CrossRef]

Salvador, P.

P. Salvador, A. Pacheco, and R. Valadas, "Modeling IP traffic: joint characterization of packet arrivals and packet sizes using BMAPs," Elsevier J. Comput. Networks 44-3, 335-352 (2004).
[CrossRef]

Simeonidou, D.

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, "The application of optical packet switching in future communication networks," IEEE Commun. Mag. 3, 128-135 (2001).
[CrossRef]

Tamil, L.

L. Tancevski, S. Yegnanarayanan, G. Castanon, L. Tamil, F. Masetti, and T. McDermott, "Optical routing of asynchronous, variable length packets," IEEE J. Sel. Areas Commun. 18-10, 2084-2093 (2000).

Tancevski, L.

L. Tancevski, S. Yegnanarayanan, G. Castanon, L. Tamil, F. Masetti, and T. McDermott, "Optical routing of asynchronous, variable length packets," IEEE J. Sel. Areas Commun. 18-10, 2084-2093 (2000).

Tetali, P.

P. D. BergstromJr., M. A. Ingram, A. J. Vernon, J. L. A. Hughes, and P. Tetali, "A Markov chain model for an Optical Shared-Memory Packet Switch," IEEE Trans. Commun. 47-10, 1593-1603 (1999).
[CrossRef]

Tucker, R. S.

R. S. Tucker, "The role of optics and electronics in high capacity routers," IEEE/OSA J. Lightwave Technol. 24-12, 4655-4673 (2006).
[CrossRef]

R. S. Tucker et al., "Slow-light optical buffers: capabilities and fundamental limitations," IEEE/OSA J. Lightwave Technol. 23-12, 4046-4066 (2005).
[CrossRef]

Tzanakaki, A.

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, "The application of optical packet switching in future communication networks," IEEE Commun. Mag. 3, 128-135 (2001).
[CrossRef]

Valadas, R.

P. Salvador, A. Pacheco, and R. Valadas, "Modeling IP traffic: joint characterization of packet arrivals and packet sizes using BMAPs," Elsevier J. Comput. Networks 44-3, 335-352 (2004).
[CrossRef]

Vernon, A. J.

P. D. BergstromJr., M. A. Ingram, A. J. Vernon, J. L. A. Hughes, and P. Tetali, "A Markov chain model for an Optical Shared-Memory Packet Switch," IEEE Trans. Commun. 47-10, 1593-1603 (1999).
[CrossRef]

Waldman, H.

R. C. Almeida, J. U. Pelegrini, and H. Waldman, "A generic-traffic optical buffer modeling for asynchronous optical switching networks," IEEE Commun. Lett. 9-2, 175 - 177 (2005).
[CrossRef]

Yegnanarayanan, S.

L. Tancevski, S. Yegnanarayanan, G. Castanon, L. Tamil, F. Masetti, and T. McDermott, "Optical routing of asynchronous, variable length packets," IEEE J. Sel. Areas Commun. 18-10, 2084-2093 (2000).

Yeo, Y. K.

Y. K. Yeo et al., "A dynamically reconfigurable folded-path time delay buffer for optical packet switching," IEEE Photon. Technol. Lett. 16-11, 2559-2561 (2004).
[CrossRef]

Yoo, S.

S. Yoo and B. Mukherjee, "Advances in photonic packet switching an overview," IEEE Commun. Mag. 2, 84-94 (2000).

Zhang, T.

T. Zhang, K. Lu, and J. R. Jue, "Shared fiber delay line buffers in asynchronous optical packet switches," IEEE J. Sel. Areas Commun. 24-4, 118-127 (2006).
[CrossRef]

Zhu, X.

X. Zhu and J. M. Khan "Queuing models of optical delay lines in synchronous and asynchronous optical packet-switching networks,' Optical Engin. 42-6, 1741-1748 (2003).
[CrossRef]

ACM SIGCOMM Computer Communication Review (1)

N. McKeown et al., "Part III: routers with very small buffers," ACM SIGCOMM Computer Communication Review 35-2, 73-89 (2005).
[CrossRef]

IEE Proc. Optoelectron (1)

S. H. Chin, A. Franzen, D. K. Hunter, and I. Andonovic, "Synchronisation schemes for optical networks," IEE Proc. Optoelectron 147-6, 423-427 (2000).
[CrossRef]

IEEE Commun. Lett. (2)

F. Callegati, "Optical buffers for variable length packets," IEEE Commun. Lett. 4-9, 292 - 294 (2000).
[CrossRef]

R. C. Almeida, J. U. Pelegrini, and H. Waldman, "A generic-traffic optical buffer modeling for asynchronous optical switching networks," IEEE Commun. Lett. 9-2, 175 - 177 (2005).
[CrossRef]

IEEE Commun. Mag. (4)

S. Yoo and B. Mukherjee, "Advances in photonic packet switching an overview," IEEE Commun. Mag. 2, 84-94 (2000).

D. K. Hunter and I. Andonovic, "Approaches to optical internet packet switching," IEEE Commun. Mag. 9, 116-122 (2000).
[CrossRef]

M. J. O’Mahony, D. Simeonidou, D. K. Hunter, and A. Tzanakaki, "The application of optical packet switching in future communication networks," IEEE Commun. Mag. 3, 128-135 (2001).
[CrossRef]

D. K. Hunter et al., "WASPNET: a wavelength switched packet network," IEEE Commun. Mag. 3, 84-94 (1999).

IEEE J. Sel. Areas Commun. (4)

I. Chlamtac et al, "CORD: contention resolution by delay lines," IEEE J. Sel. Areas Commun. 14-5, 1014 - 1029 (1996).
[CrossRef]

P. Gambini et al., "Transparent optical packet switching: network architecture and demonstration in KEOPS project," IEEE J. Sel. Areas Commun. 16-17, 1245 - 1259 (1998).
[CrossRef]

L. Tancevski, S. Yegnanarayanan, G. Castanon, L. Tamil, F. Masetti, and T. McDermott, "Optical routing of asynchronous, variable length packets," IEEE J. Sel. Areas Commun. 18-10, 2084-2093 (2000).

T. Zhang, K. Lu, and J. R. Jue, "Shared fiber delay line buffers in asynchronous optical packet switches," IEEE J. Sel. Areas Commun. 24-4, 118-127 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Y. K. Yeo et al., "A dynamically reconfigurable folded-path time delay buffer for optical packet switching," IEEE Photon. Technol. Lett. 16-11, 2559-2561 (2004).
[CrossRef]

IEEE Trans. Commun. (2)

M. J. Karol et al., "Input versus output queueing on a space-division packet switch," IEEE Trans. Commun. 35-12, 1347-1356 (1987).
[CrossRef]

P. D. BergstromJr., M. A. Ingram, A. J. Vernon, J. L. A. Hughes, and P. Tetali, "A Markov chain model for an Optical Shared-Memory Packet Switch," IEEE Trans. Commun. 47-10, 1593-1603 (1999).
[CrossRef]

IEEE/ACM Trans. on Networking (1)

V. Paxson and S. Floyd, "Wide-Area Traffic: The Failure of Poisson Modeling," IEEE/ACM Trans. on Networking,  3-3, 226-244 (1995).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (4)

M. C. Chia, D. K. Hunter et al., "Packet loss and delay performance of feedback and feed-forward arrayed-waveguide gratings-based optical packet switches with WDM inputs-outputs," IEEE/OSA J. Lightwave Technol. 19-9, 1241 - 1254 (2001).
[CrossRef]

D. K. Hunter, M. C. Chia, and I. Andonovic, "Buffering in optical packet switches," IEEE/OSA J. Lightwave Technol. 16-12, 2081-1094 (1998).
[CrossRef]

R. S. Tucker et al., "Slow-light optical buffers: capabilities and fundamental limitations," IEEE/OSA J. Lightwave Technol. 23-12, 4046-4066 (2005).
[CrossRef]

R. S. Tucker, "The role of optics and electronics in high capacity routers," IEEE/OSA J. Lightwave Technol. 24-12, 4655-4673 (2006).
[CrossRef]

Optical Engin. (1)

X. Zhu and J. M. Khan "Queuing models of optical delay lines in synchronous and asynchronous optical packet-switching networks,' Optical Engin. 42-6, 1741-1748 (2003).
[CrossRef]

Other (10)

CISCO router specification, "CRS1 specification" (CISCO Networks 2004), http://newsroom.cisco.com/dlls/2004/prod_052504.html.

F. Callegati, "On the design of optical buffers for variable length packet traffic," Ninth International Conference on Computer Communications and Networks (Las Vegas, Nevada, USA, 6-18 Oct. 2000), pp. 448 - 452.

R. Geldenhuys et al., "Selecting fibre delay line distributions for travelling buffers in an all-optical packet switched cross-connect," in Proceedings of IEEE CCECE (Montreal, Canada, 4-7 May 2003).

J. Cao, W. Cleveland, D. Lin, and D. Sun, "Internet traffic tends toward Poisson and independent as the load increases" in Nonlinear Estimation and Classification (Springer, New York 2002).

J. Cao, W. Cleveland, D. Lin, and D. Sun, "The effect of statistical multiplexing on the long range dependence of Internet packet traffic," Bell Labs Tech. Reports (2002).

P. Salvador, A. Pacheco, and R. Valadas, "Modeling IP traffic: joint characterization of packet arrivals and packet sizes using BMAPs," Elsevier J. Comput. Networks 44-3, 335-352 (2004).
[CrossRef]

R. C. Almeida, J. U. Pelegrini, and H. Waldman, "Optical buffer modeling for performance evaluation considering any packet inter-arrival time distribution," IEEE International Conference on Communications3, (Paris, France, 20-24 June 2004), pp. 1771 - 1775.

G. F. Lawler, Introduction to Stochastic Processes Chapman & Hall (1995).

L. Kleinrock, Queuing systems: volume I - Theory (Wiley Interscience, New York, 1975).

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Figures (13)

Fig. 1.
Fig. 1.

Different buffering elements using ODLs

Fig. 2.
Fig. 2.

A typical optical buffer array with ODLs: (a) Switch (b) Buffer

Fig. 3.
Fig. 3.

The new MISO-FIFO buffer structure with fixed length ODLs

Fig. 4.
Fig. 4.

An example timing diagram for a MISO-FIFO buffer with fixed length ODLs

Fig. 5.
Fig. 5.

Markov chain state transitions for the MISO-FIFO buffer with fixed length ODLs

Fig. 6.
Fig. 6.

Timing diagram when packets are discarded

Fig. 7.
Fig. 7.

Modified Markov chain state transitions for the MISO-FIFO buffer with fixed length ODLs

Fig 8.
Fig 8.

An example (a) MISO-FIFO buffer with CFDLs (F=2) and (b) the timing diagram for MISO-FIFO buffer with CFDLs (F=2)

Fig 9.
Fig 9.

Markov chain state transition for the MISO-FIFO buffer with CFDLs

Fig. 10.
Fig. 10.

Output link utilization vs. buffer size for different traffic loads in the MISO-FIFO buffer with fixed ODLs

Fig. 11.
Fig. 11.

Output link utilization vs. buffer size for traffic load=0.8 and 1.0 in the MISO-FIFO buffer with CFDLs

Fig. 12.
Fig. 12.

Comparison of analytical and simulation model for the MISO-FIFO buffer with CFDLs for F=2, 4

Fig. 13.
Fig. 13.

Comparison of analytical and simulation models for packet loss probability (PLP) for the MISO-FIFO buffer using CFDLs with F=2

Equations (17)

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π i = { j = 1 B + 1 exp ( jQ ) π j i = 1 j = i 1 B + 1 { exp [ ( j i + 1 ) ] exp [ ( j i + 2 ) ] } π j 2 i B + 1 }
π i = q i 1 ( 1 q ) 1 q B + 1 1 i B + 1
PLP = π B + 1 ( 1 e Q ) = ( 1 q ) q B + 1 ( 1 + q ) ( 1 q B + 1 )
ρ = Q Q . PLP = Q ( 1 PLP ) .
Lim q 1 Lim B ρ = Lim q 1 Lim B Q [ 1 ( q 1 ) q B + 1 ( 1 + q ) ( q B + 1 1 ) ] = Q
p ( D 2 ) = p ( D 2 inter-arival time D inter-arival time D )
= D 2 D λ e λ t dt 0 D λ e λ t dt = e Q 2 [ 1 e Q 2 1 e Q ] = x where x = e Q 2 [ 1 e Q 2 1 e Q ]
p ( D ) = p ( inter-arival time < D 2 inter-arival time D )
= 1 p ( D 2 ) = [ 1 e Q 2 1 e Q ] = xy where y = e Q 2
π i = { j = 1 B + 1 p j , i π j i = 1 j = i 1 B + 1 p j , i π j 2 i B + 1 }
p n , r = i = 0 ( n r ) p ( i ) [ p ( D 2 ) e ( n r + i + 2 ) Q 2 ( 1 e Q 2 )
+ p ( D ) e ( n r + i + 2 ) Q 2 ( 1 e Q ) ] where p ( i ) = ( n r ) C i y i x n r
= ( 2 e Q 1 + e Q 2 ) n r + 1 [ e Q 2 2 e Q + 1 2 ]
p r , 1 = i = 0 ( r 1 ) p ( i ) e ( r 1 + i + 2 ) Q 2 where p ( i ) = ( r 1 ) C i y i x r 1
= ( 2 e Q 1 + e Q 2 ) r 1 e Q
p r , r = p ( D 2 ) ( e Q e 3 Q 2 ) + p ( D ) ( e Q e 2 Q )
= e Q 1 + e Q 2 ( 1 + e Q 2 2 e Q )

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