# HOMOTHETIC APPROXIMATIONS FOR STOCHASTIC PN

### Dimitri Lefebvre

#### Abstract

Reliability analysis is often based on stochastic discrete event models like 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. For this reason, fluidification is an interesting alternative to estimate the asymptotic behaviour of stochastic processes with continuous Petri nets. Unfortunately, the asymptotic mean marking of stochastic and continuous Petri nets are mainly often different. This paper proposes a geometric approach that leads to a homothetic approximation of the stochastic steady state in specific regions of the marking space.

#### References

- Bobbio A., Puliafito A., Telek M., Trivedi K. (1998) Recent Developments in Stochastic Petri Nets, J. of Cir., Syst., and Comp., Vol. 8, No. 1, pp. 119--158.
- David R., Alla H., (1992) Petri nets and grafcet - tools for modelling DES, Prentice Hall, London.
- JĂșlvez G., Recalde L. Silva M. (2005) Steady-state performance evaluation of continuous mono-Tsemiflow Petri nets, Automatica, 41 (4), pp. 605-616.
- Lefebvre D., Leclercq E., Khalij L., Souza de Cursi E., El Akchioui N. (2009), 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.
- Lefebvre D., Leclercq E., (2010), Approximation of Stochastic Petri Nets steady state by CPNs with constant and piecewise-constant maximal firing speeds, MOSIM 2010, Hammamet, Tunisia.
- Molloy M.K., (1982) Performance analysis using stochastic Petri nets, IEEE Tran. on Computers C, vol. 31, pp. 913 - 917.
- Rausand M. and Hoyland A. (2004), System reliability theory: models, statistical methods, and applications, Wiley, Hoboken, New Jersey.
- Silva M. and Recalde L. (2004) On fluidification of Petri Nets: from discrete to hybrid and continuous models, An. Reviews in Control, Vol. 28, no. 2, pp. 253-266.
- Vazquez R., Recalde L., Silva M., (2008) Stochastic continuous-state approximation of markovian Petri net systems, Proc. IEEE - CDC08, pp. 901 - 906, Cancun, Mexico.

#### Paper Citation

#### in Harvard Style

Lefebvre D. (2010). **HOMOTHETIC APPROXIMATIONS FOR STOCHASTIC PN** . In *Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,* ISBN 978-989-8425-01-0, pages 364-367. DOI: 10.5220/0002918003640367

#### in Bibtex Style

@conference{icinco10,

author={Dimitri Lefebvre},

title={HOMOTHETIC APPROXIMATIONS FOR STOCHASTIC PN},

booktitle={Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,},

year={2010},

pages={364-367},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002918003640367},

isbn={978-989-8425-01-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,

TI - HOMOTHETIC APPROXIMATIONS FOR STOCHASTIC PN

SN - 978-989-8425-01-0

AU - Lefebvre D.

PY - 2010

SP - 364

EP - 367

DO - 10.5220/0002918003640367