Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net

Nabil El Akchioui

doi:doi:10.11648/j.ajesa.20170504.11

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Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net

Nabil El Akchioui

Received: 6 November 2017 Accepted: 16 November 2017 Published: 2 January 2018

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Abstract

Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. Moreover, the convergence of stochastic estimators is slow. For these reasons, fluidification can be investigated to estimate the asymptotic behaviour of stochastic processes with timed continuous Petri nets. The contributions of this paper are to sum up some properties of the asymptotic mean marking and average throughputs of stochastic and timed continuous Petri nets, then to point out the limits of the fluidification in the context of the stochastic steady state approximation. To overcome these limitations, the new semantic and the condition for convergence is proposed: fluid Petri nets with Non Linear Timed Continuous Petri Net (NL-CPN).

DOI	10.11648/j.ajesa.20170504.11
Published in	American Journal of Embedded Systems and Applications (Volume 5, Issue 4, July 2017)
Page(s)	29-34
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), 2024. Published by Science Publishing Group

Keywords

Fluidification, Stochastic Petri Nets, Continuous Petri Nets, Steady State, Reliability Analysis

References

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[3]	Campos J., Chiola G., Silva M Ergodicity and throughut bounds of Petri nets with unique consistent firing count vector, IEEE Transactions on Software Engineering, Vol. 17, no 2, pp. 117-125, 1991.
[4]	David R., Alla H., Petri nets and grafcet – tools for modelling discrete events systems, Prentice Hall, London, 1992.
[5]	Diaz M., (Les réseaux de Petri: modèles fondamentaux, Hermes, Paris, 2001.
[6]	Júlvez G., Recalde L. Silva M. Steady-state performance evaluation of continuous mono-T- semiflow Petri nets, Automatica, Vol. 41, no. 4, pp. 605-616, 2005.
[7]	Mahulea C., Ramirez Trevino A., Recalde L., Silva M., Steady state control reference and token conservation laws in continuous Petri nets, Trans. IEEE – TASE, Vol. 5, no. 2, pp. 307–320, 2008.
[8]	Molloy M. K. On the integration of delay and troughput in distributed processing models, Ph. D, UCLA, Los Angeles, USA, 1981.
[9]	Molloy M. K., Performance analysis using stochastic Petri nets, IEEE Transactions on Computers C, vol. 31, pp. 913–917, 1982.
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[11]	Silva M. and Recalde L. Petri nets and integrality relaxations: a view of continuous Petri nets, Trans. IEEE – SMC, part C, Vol. 32, no. 4, pp. 314-326, 2002.
[12]	Silva M. and Recalde L. On fluidification of Petri Nets: from discrete to hybrid and continuous models, Annual Reviews in Control, Vol. 28, no. 2, pp. 253-266, 2004.
[13]	Vazquez R., Recalde L., Silva M., Stochastic continuous-state approximation of markovian Petri net systems, Proceeding IEEE – CDC08, pp. 901–906, Cancun, Mexico, 2008.
[14]	Vazquez R., Silva M., (Hybrid Approximations of Markovian Petri Nets, Proceeding IFAC – ADHS, pp. 56-61, Zaragoza, Spain, 2009.
[15]	Lefebvre D, E. Leclercq, L. Khalij, E. Souza de Cursi, N. El Akchioui, Approximation of MTS stochastic Petri nets steady state by means of continuous Petri nets: a numerical approach, Proc. IFAC ADHS, pp. 62-67, Zaragoza, Spain, 2009.
[16]	Lefebvre D, E. Leclercq, Piecewise constant timed continuous PNs for the steady state estimation of stochastic PNs, DISC, DOI: 10.1007/s10626-011-0114-y, 2011.
[17]	Lefebvre D, E. Leclercq, N. El Akchioui, L. Khalij, E. Souza de Cursi, A geometric approach for the homothetic approximation of stochastic Petri nets, Proc. IFAC WODES, Berlin, Germany, 2010.
[18]	Lefebvre D, About the stochastic and continuous Petri nets equivalence in long run, Non-Linear Analysis, Hybrid Systems (NAHS), vol. 5, pp. 394-406, 2011.
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Nabil El Akchioui. (2018). Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net. American Journal of Embedded Systems and Applications, 5(4), 29-34. https://doi.org/10.11648/j.ajesa.20170504.11

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ACS Style

Nabil El Akchioui. Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net. Am. J. Embed. Syst. Appl. 2018, 5(4), 29-34. doi: 10.11648/j.ajesa.20170504.11

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AMA Style

Nabil El Akchioui. Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net. Am J Embed Syst Appl. 2018;5(4):29-34. doi: 10.11648/j.ajesa.20170504.11

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@article{10.11648/j.ajesa.20170504.11,
  author = {Nabil El Akchioui},
  title = {Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net},
  journal = {American Journal of Embedded Systems and Applications},
  volume = {5},
  number = {4},
  pages = {29-34},
  doi = {10.11648/j.ajesa.20170504.11},
  url = {https://doi.org/10.11648/j.ajesa.20170504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajesa.20170504.11},
  abstract = {Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. Moreover, the convergence of stochastic estimators is slow. For these reasons, fluidification can be investigated to estimate the asymptotic behaviour of stochastic processes with timed continuous Petri nets. The contributions of this paper are to sum up some properties of the asymptotic mean marking and average throughputs of stochastic and timed continuous Petri nets, then to point out the limits of the fluidification in the context of the stochastic steady state approximation. To overcome these limitations, the new semantic and the condition for convergence is proposed: fluid Petri nets with Non Linear Timed Continuous Petri Net (NL-CPN).},
 year = {2018}
}

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TY  - JOUR
T1  - Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net
AU  - Nabil El Akchioui
Y1  - 2018/01/02
PY  - 2018
N1  - https://doi.org/10.11648/j.ajesa.20170504.11
DO  - 10.11648/j.ajesa.20170504.11
T2  - American Journal of Embedded Systems and Applications
JF  - American Journal of Embedded Systems and Applications
JO  - American Journal of Embedded Systems and Applications
SP  - 29
EP  - 34
PB  - Science Publishing Group
SN  - 2376-6085
UR  - https://doi.org/10.11648/j.ajesa.20170504.11
AB  - Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. Moreover, the convergence of stochastic estimators is slow. For these reasons, fluidification can be investigated to estimate the asymptotic behaviour of stochastic processes with timed continuous Petri nets. The contributions of this paper are to sum up some properties of the asymptotic mean marking and average throughputs of stochastic and timed continuous Petri nets, then to point out the limits of the fluidification in the context of the stochastic steady state approximation. To overcome these limitations, the new semantic and the condition for convergence is proposed: fluid Petri nets with Non Linear Timed Continuous Petri Net (NL-CPN).
VL  - 5
IS  - 4
ER  -

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Author Information

Nabil El Akchioui

Faculty of Science and Technology, University the First Mohamed, Oujda, Morocco

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Nabil El Akchioui. (2018). Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net. American Journal of Embedded Systems and Applications, 5(4), 29-34. https://doi.org/10.11648/j.ajesa.20170504.11

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ACS Style

Nabil El Akchioui. Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net. Am. J. Embed. Syst. Appl. 2018, 5(4), 29-34. doi: 10.11648/j.ajesa.20170504.11

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AMA Style

Nabil El Akchioui. Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net. Am J Embed Syst Appl. 2018;5(4):29-34. doi: 10.11648/j.ajesa.20170504.11

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@article{10.11648/j.ajesa.20170504.11,
  author = {Nabil El Akchioui},
  title = {Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net},
  journal = {American Journal of Embedded Systems and Applications},
  volume = {5},
  number = {4},
  pages = {29-34},
  doi = {10.11648/j.ajesa.20170504.11},
  url = {https://doi.org/10.11648/j.ajesa.20170504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajesa.20170504.11},
  abstract = {Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. Moreover, the convergence of stochastic estimators is slow. For these reasons, fluidification can be investigated to estimate the asymptotic behaviour of stochastic processes with timed continuous Petri nets. The contributions of this paper are to sum up some properties of the asymptotic mean marking and average throughputs of stochastic and timed continuous Petri nets, then to point out the limits of the fluidification in the context of the stochastic steady state approximation. To overcome these limitations, the new semantic and the condition for convergence is proposed: fluid Petri nets with Non Linear Timed Continuous Petri Net (NL-CPN).},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Fluidification of Stochastic Petri Net by Non Linear Timed Continuous Petri Net
AU  - Nabil El Akchioui
Y1  - 2018/01/02
PY  - 2018
N1  - https://doi.org/10.11648/j.ajesa.20170504.11
DO  - 10.11648/j.ajesa.20170504.11
T2  - American Journal of Embedded Systems and Applications
JF  - American Journal of Embedded Systems and Applications
JO  - American Journal of Embedded Systems and Applications
SP  - 29
EP  - 34
PB  - Science Publishing Group
SN  - 2376-6085
UR  - https://doi.org/10.11648/j.ajesa.20170504.11
AB  - Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. Moreover, the convergence of stochastic estimators is slow. For these reasons, fluidification can be investigated to estimate the asymptotic behaviour of stochastic processes with timed continuous Petri nets. The contributions of this paper are to sum up some properties of the asymptotic mean marking and average throughputs of stochastic and timed continuous Petri nets, then to point out the limits of the fluidification in the context of the stochastic steady state approximation. To overcome these limitations, the new semantic and the condition for convergence is proposed: fluid Petri nets with Non Linear Timed Continuous Petri Net (NL-CPN).
VL  - 5
IS  - 4
ER  -

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