Production Networks with Stochastic Machinery Default

  • Simone Göttlich
  • Stephan Martin
  • Thorsten Sickenberger
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 17)

Abstract

We present a model of production networks that includes random breakdowns of individual processors. The defaults of processors are exponentially distributed and the time-continuous formulation of network dynamics yields a coupled PDE-ODE system with Markovian switching. Its solution is a piecewise deterministic process, which allows us to use a modified stochastic simulation algorithm to trace stochastic events and to simulate the deterministic behavior of the network between them. The impact of stochastic default is illustrated with an exemplary Monte-Carlo simulation.

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References

  1. 1.
    Davis, M.H.A.: Markov Models and Optimisation. Monograph on Statistics and Applied Probability 49, Chapmand & Hall, London (1993)Google Scholar
  2. 2.
    D’Apice, C., Göttlich, S., Herty, M., Piccoli, B.: Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach. SIAM (2010)Google Scholar
  3. 3.
    Degond, P., Ringhofer, C.: Stochastic dynamics of long supply chains with random breakdowns. SIAM J. Appl. Math. 68(1), 59–79 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Phys. Chem. A 104, 403–434 (1976)MathSciNetGoogle Scholar
  5. 5.
    Göttlich, S, Herty, M., Klar, A.: Network models for supply chains. Comm. Math. Sci. 3(4), 545–559 (2005)Google Scholar
  6. 6.
    Göttlich, S., Martin, S., Sickenberger, T.: Time-continuous production networks with random breakdowns. Networks and Heterogeneous Media (NHM) 6(4), 695–714 (2011) DOI: 10.3934/nhm.2011.6.695Google Scholar
  7. 7.
    Kelly, F.P., Zachary, S., Ziedins, I. (eds.): Stochastic Networks: Theory and Applications. Oxford University Press, Oxford (2002)Google Scholar
  8. 8.
    Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Simone Göttlich
    • 1
  • Stephan Martin
    • 2
  • Thorsten Sickenberger
    • 3
  1. 1.School of Business Informatics and MathematicsUniversity of MannheimMannheimGermany
  2. 2.Department of MathematicsTU KaiserslauternKaiserslauternGermany
  3. 3.Department of MathematicsMaxwell Institute and Heriot-Watt UniversityEdinburghUK

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