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.

Keywords

Supply Chain Switching Point Production Network Product Density Markovian Switching 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations