Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions

Yasuji Murakami

doi:doi:10.11648/j.ajam.s.2014020601.11

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Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions

Yasuji Murakami

Published in American Journal of Applied Mathematics (Volume 2, Issue 6-1)

Received: 19 October 2014 Accepted: 28 November 2014 Published: 10 January 2015

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Abstract

Asynchronous optical packet switching seems to be suitable as a transport technology for the next-generation Internet due to the variable lengths of IP packets. Optical buffers in the output port are an integral part for solving contention by exploiting the time domain. Fiber delay lines (FDLs) are a well-known technique for achieving optical buffers. This work aims to give a highly accurate approximation of the blocking probabilities of the optical buffers for a generally distributed packet length even when the offered load is extremely low. Such a tool is needed for investigating and designing realistic optical packet switches, which will be used for low-offered-load and low-packet-loss optical IP networks. We use the asymptotic expansion for the decay rate, resulting in a highly accurate approximation. By using the fourth order approximation of the decay rate, an accuracy within 10 % was obtained for both the exponential and uniform distribution cases of an offered load greater than 0.3. The approximations established in this work can be applied to investigate multiclass optical buffers for priority queueing.

Published in	American Journal of Applied Mathematics (Volume 2, Issue 6-1) This article belongs to the Special Issue Switched Dynamics with Its Applications
DOI	10.11648/j.ajam.s.2014020601.11
Page(s)	1-10
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Asynchronous Optical Switching, Optical Buffers, Blocking Probabilities, General Packet-Length Distributions

References

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[2]	R. S Tucker, “Scalability and energy consumption of optical and electronic packet switching,” J. Lightwave Technol., Vol. 29, No. 16, pp. 2410-2421, 2011.
[3]	F. Callegati, “Optical buffers for variable length packets,” IEEE Commun. Lett., Vol. 4, No. 9, pp. 292-294, 2000.
[4]	R. C. Almeida, J. U. Pelegrini, and H. Waldman, “A generic-traffic optical buffer modeling for asynchronous optical switching networks,” IEEE Commun. Lett., Vol. 3, No. 2, pp. 175-177, 2005.
[5]	X. Ma, “Modeling and design of WDM optical buffers in asynchronous and variable-length optical packets switches,” Optical Commun., No. 269, pp. 53-63, 2007.
[6]	J. Liu, T. T. Lee, X. Jiang, and 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.
[7]	H. E. Kankaya and N. Akar, “Exact analysis of single-wavelength optical buffers with feedback Markov fluid queues,” J. Opt. Commun. Netw., Vol. 1, No. 6, pp. 530-542, 2009.
[8]	W. Rogiest, and H. Bruneel, “Exact optimization method for an FDL buffer with variable packet length”, Photon. Technol. Lett., Vol. 22, No. 4, pp. 242-244, 2010.
[9]	Y. Murakami, “An approximation for blocking probabilities and delays of optical buffer with general packet-length distributions,” J. Lightwave Technol., Vol. 30, No. 1, pp. 54-66, 2012.
[10]	W. Rogiest et al., “Heuristic performance model of optical buffers for variable length packets,” Photon Netw. Commun., Vol. 26, pp. 65-73, 2013.
[11]	ITU-T Recommendation Y.1541 (12/2011), Network performance objectives for IP-based services.
[12]	L. Kleinrock, “Queueing systems, Vol. 1: Theory”, John Wiley & Sons, New York, 1975.
[13]	A. A. Fredricks, “A class of approximations for the waiting time distribution in a GI/G/1 queueing system,” Bell Syst. Tech. J. Vol. 61, pp. 295-325, 1982.
[14]	G. L. Choudhury, and W. Whitt, “Heavy-traffic asymptotic expansions for the asymptotic decay rates in the BMAP/G/1 queue,” Stochastic Models, Vol. 10, No. 2, pp. 453-498, 1994.
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[16]	see 3.6.25 in M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions, 10th printing,” National Bureaus of Standards, U. S. Government Printing Office, Washington, D. C., 1972.
[17]	F. Xue et al., “Design and experimental demonstration of a variable-length optical packet routing system with unified contention resolution,” J. Lightwave Technol., vol. 22, no. 11, pp. 2570-2581, 2004.
[18]	The cooperative association for the internet data analysis – Packet size distribution comparison between internet links in 1998 and 2008, www.caida.org/research/traffic-analysis/ , 2008.
[19]	E. R. S. Castro, M. S. Alencar and I. E. Fonseca, “Probability density functions of the packet length for computer networks with bimodal traffic,” International J. of Computer Networks & Communication, vol. 5, no. 3, pp.17-31, 2013.

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APA Style

Yasuji Murakami. (2015). Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions. American Journal of Applied Mathematics, 2(6-1), 1-10. https://doi.org/10.11648/j.ajam.s.2014020601.11

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ACS Style

Yasuji Murakami. Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions. Am. J. Appl. Math. 2015, 2(6-1), 1-10. doi: 10.11648/j.ajam.s.2014020601.11

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AMA Style

Yasuji Murakami. Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions. Am J Appl Math. 2015;2(6-1):1-10. doi: 10.11648/j.ajam.s.2014020601.11

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@article{10.11648/j.ajam.s.2014020601.11,
  author = {Yasuji Murakami},
  title = {Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {6-1},
  pages = {1-10},
  doi = {10.11648/j.ajam.s.2014020601.11},
  url = {https://doi.org/10.11648/j.ajam.s.2014020601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.s.2014020601.11},
  abstract = {Asynchronous optical packet switching seems to be suitable as a transport technology for the next-generation Internet due to the variable lengths of IP packets. Optical buffers in the output port are an integral part for solving contention by exploiting the time domain. Fiber delay lines (FDLs) are a well-known technique for achieving optical buffers. This work aims to give a highly accurate approximation of the blocking probabilities of the optical buffers for a generally distributed packet length even when the offered load is extremely low. Such a tool is needed for investigating and designing realistic optical packet switches, which will be used for low-offered-load and low-packet-loss optical IP networks. We use the asymptotic expansion for the decay rate, resulting in a highly accurate approximation. By using the fourth order approximation of the decay rate, an accuracy within 10 % was obtained for both the exponential and uniform distribution cases of an offered load greater than 0.3. The approximations established in this work can be applied to investigate multiclass optical buffers for priority queueing.},
 year = {2015}
}

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TY  - JOUR
T1  - Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions
AU  - Yasuji Murakami
Y1  - 2015/01/10
PY  - 2015
N1  - https://doi.org/10.11648/j.ajam.s.2014020601.11
DO  - 10.11648/j.ajam.s.2014020601.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 1
EP  - 10
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.s.2014020601.11
AB  - Asynchronous optical packet switching seems to be suitable as a transport technology for the next-generation Internet due to the variable lengths of IP packets. Optical buffers in the output port are an integral part for solving contention by exploiting the time domain. Fiber delay lines (FDLs) are a well-known technique for achieving optical buffers. This work aims to give a highly accurate approximation of the blocking probabilities of the optical buffers for a generally distributed packet length even when the offered load is extremely low. Such a tool is needed for investigating and designing realistic optical packet switches, which will be used for low-offered-load and low-packet-loss optical IP networks. We use the asymptotic expansion for the decay rate, resulting in a highly accurate approximation. By using the fourth order approximation of the decay rate, an accuracy within 10 % was obtained for both the exponential and uniform distribution cases of an offered load greater than 0.3. The approximations established in this work can be applied to investigate multiclass optical buffers for priority queueing.
VL  - 2
IS  - 6-1
ER  -

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Author Information

Yasuji Murakami

The department of Telecommunications and Computer Networks, Osaka Electro-Communication University, Osaka, Japan

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APA Style

Yasuji Murakami. (2015). Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions. American Journal of Applied Mathematics, 2(6-1), 1-10. https://doi.org/10.11648/j.ajam.s.2014020601.11

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ACS Style

Yasuji Murakami. Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions. Am. J. Appl. Math. 2015, 2(6-1), 1-10. doi: 10.11648/j.ajam.s.2014020601.11

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AMA Style

Yasuji Murakami. Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions. Am J Appl Math. 2015;2(6-1):1-10. doi: 10.11648/j.ajam.s.2014020601.11

Copy | Download

@article{10.11648/j.ajam.s.2014020601.11,
  author = {Yasuji Murakami},
  title = {Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {6-1},
  pages = {1-10},
  doi = {10.11648/j.ajam.s.2014020601.11},
  url = {https://doi.org/10.11648/j.ajam.s.2014020601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.s.2014020601.11},
  abstract = {Asynchronous optical packet switching seems to be suitable as a transport technology for the next-generation Internet due to the variable lengths of IP packets. Optical buffers in the output port are an integral part for solving contention by exploiting the time domain. Fiber delay lines (FDLs) are a well-known technique for achieving optical buffers. This work aims to give a highly accurate approximation of the blocking probabilities of the optical buffers for a generally distributed packet length even when the offered load is extremely low. Such a tool is needed for investigating and designing realistic optical packet switches, which will be used for low-offered-load and low-packet-loss optical IP networks. We use the asymptotic expansion for the decay rate, resulting in a highly accurate approximation. By using the fourth order approximation of the decay rate, an accuracy within 10 % was obtained for both the exponential and uniform distribution cases of an offered load greater than 0.3. The approximations established in this work can be applied to investigate multiclass optical buffers for priority queueing.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Asymptotic Analysis for Blocking Probabilities of Optical Buffer with General Packet-Length Distributions
AU  - Yasuji Murakami
Y1  - 2015/01/10
PY  - 2015
N1  - https://doi.org/10.11648/j.ajam.s.2014020601.11
DO  - 10.11648/j.ajam.s.2014020601.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 1
EP  - 10
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.s.2014020601.11
AB  - Asynchronous optical packet switching seems to be suitable as a transport technology for the next-generation Internet due to the variable lengths of IP packets. Optical buffers in the output port are an integral part for solving contention by exploiting the time domain. Fiber delay lines (FDLs) are a well-known technique for achieving optical buffers. This work aims to give a highly accurate approximation of the blocking probabilities of the optical buffers for a generally distributed packet length even when the offered load is extremely low. Such a tool is needed for investigating and designing realistic optical packet switches, which will be used for low-offered-load and low-packet-loss optical IP networks. We use the asymptotic expansion for the decay rate, resulting in a highly accurate approximation. By using the fourth order approximation of the decay rate, an accuracy within 10 % was obtained for both the exponential and uniform distribution cases of an offered load greater than 0.3. The approximations established in this work can be applied to investigate multiclass optical buffers for priority queueing.
VL  - 2
IS  - 6-1
ER  -

Copy | Download