American Institute of Mathematical Sciences

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December  2011, 6(4): 695-714. doi: 10.3934/nhm.2011.6.695

Time-continuous production networks with random breakdowns

Simone Göttlich 1, , Stephan Martin 2, and  Thorsten Sickenberger 3,

1. 

Department of Mathematics, University of Mannheim, D-68131 Mannheim
2. 

Dept. of Mathematics, TU Kaiserslautern, 67663 Kaiserslautern, Germany
3. 

Maxwell Institute and Heriot-Watt University, Dept. of Mathematics, Edinburgh, EH14 4AS, Scotland, United Kingdom

Received  April 2011 Revised  October 2011 Published  December 2011

Our main objective is the modelling and simulation of complex production networks originally introduced in [15, 16] with random breakdowns of individual processors. Similar to [10], the breakdowns of processors are exponentially distributed. The resulting network model consists of coupled system of partial and ordinary differential equations with Markovian switching and its solution is a stochastic process. We show our model to fit into the framework of piecewise deterministic processes, which allows for a deterministic interpretation of dynamics between a multivariate two-state process. We develop an efficient algorithm with an emphasis on accurately tracing stochastic events. Numerical results are presented for three exemplary networks, including a comparison with the long-chain model proposed in [10].
Keywords: piecewise deterministic processes (PDPs)., coupled PDE-ODE systems, random breakdowns, Markovian switching, Production networks, conservation laws.
Mathematics Subject Classification: Primary: 90B15, 65C05; Secondary: 65Mx.
Citation: Simone Göttlich, Stephan Martin, Thorsten Sickenberger. Time-continuous production networks with random breakdowns. Networks & Heterogeneous Media, 2011, 6 (4) : 695-714. doi: 10.3934/nhm.2011.6.695
References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains,, SIAM J. Appl. Math., 66 (2006), 896. doi: 10.1137/040604625. Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Networks and Heterogenous Media, 1 (2006), 41. doi: 10.3934/nhm.2006.1.41. Google Scholar

[3]

M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations,, Networks and Heterogenous Media, 1 (2006), 295. doi: 10.3934/nhm.2006.1.295. Google Scholar

[4]

S. Battiston, D. Delli Gatti, M. Gallegati, B. Greenwald and J. E. Stiglitz, Credit chains and bankruptcy propagation in production networks,, J. Economic Dynamics and Control, 31 (2007), 2061. doi: 10.1016/j.jedc.2007.01.004. Google Scholar

[5]

G. Bretti, C. D'Apice, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics,, Networks and Heterogeneous Media, 2 (2007), 661. doi: 10.3934/nhm.2007.2.661. Google Scholar

[6]

G. Coclite, M. Garavello and B. Piccoli, Traffic flow on road networks,, SIAM J. Mathematical Analysis, 36 (2005), 1862. doi: 10.1137/S0036141004402683. Google Scholar

[7]

C. D'Apice and R. Manzo, A fluid-dynamic model for supply chain,, Networks and Heterogeneous Media, 1 (2006), 379. doi: 10.3934/nhm.2006.1.379. Google Scholar

[8]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models. With discussion,, J. Royal Statistical Society Ser. B, 46 (1984), 353. Google Scholar

[9]

M. H. A. Davis, "Markov Models and Optimisation,", Monograph on Statistics and Applied Probability, 49 (1993). Google Scholar

[10]

P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns,, SIAM J. Appl. Math., 68 (2007), 59. doi: 10.1137/060674302. Google Scholar

[11]

A. Fügenschuh, M. Herty and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks,, SIAM J. Optimization, 16 (2006), 1155. Google Scholar

[12]

C. W. Gardiner, "Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,'' 3rd edition,, Springer Series in Synergetics, 13 (2004). Google Scholar

[13]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions,, J. Computational Phys., 22 (1976), 403. doi: 10.1016/0021-9991(76)90041-3. Google Scholar

[14]

D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems,, J. Chem. Phys., 115 (2001), 1716. doi: 10.1063/1.1378322. Google Scholar

[15]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains,, Comm. Math. Sci., 3 (2005), 545. Google Scholar

[16]

S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks,, Comm. Math. Sci., 4 (2006), 315. Google Scholar

[17]

S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks,, European J. of Operational Research, 202 (2010), 456. doi: 10.1016/j.ejor.2009.05.028. Google Scholar

[18]

M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks,, J. Optimization Theory and Application, 126 (2005), 589. doi: 10.1007/s10957-005-5499-z. Google Scholar

[19]

D. Helbing, "Verkehrsdynamik,'', Springer Verlag, (1997). doi: 10.1007/978-3-642-59063-4. Google Scholar

[20]

D. Helbing, S. Lämmer and T. Seidel, Physics, stability and dynamics of supply chains,, Physical Review E, 70 (2004), 066116. doi: 10.1103/PhysRevE.70.066116. Google Scholar

[21]

M. Herty and A. Klar, Modeling, simulation and optimization of traffic flow networks,, SIAM J. Scientific Computing, 25 (2003), 1066. doi: 10.1137/S106482750241459X. Google Scholar

[22]

M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations,, SIAM J. Mathematical Analysis, 39 (2007), 160. doi: 10.1137/060659478. Google Scholar

[23]

T. Kazangey and D. D. Sworder, Effective federal policies for regulating residential housing,, Proc. Summer Computer Simulation Conf., (1971), 1120. Google Scholar

[24]

F. P. Kelly, S. Zachary and I. Ziedins, eds., "Stochastic Networks: Theory and Apllications,", Oxford University Press, (2002). Google Scholar

[25]

C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal control for continuous supply network models,, Networks and Heterogenous Media, 1 (2006), 675. doi: 10.3934/nhm.2006.1.675. Google Scholar

[26]

G. Leugering and E. Schmidt, On the modelling and stabilization of flows in networks of open channels,, SIAM J. Control and Optimization, 41 (2002), 164. Google Scholar

[27]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,'', Imperial College Press, (2006). Google Scholar

[28]

M. Mariton, "Jump Linear Systems in Automatic Control,'', Marcel Dekker, (1990). Google Scholar

[29]

A. Martin, M. Möller and S. Moritz, Mixed integer models for the stationary case of gas network optimization,, Math. Programming, 105 (2006), 563. doi: 10.1007/s10107-005-0665-5. Google Scholar

[30]

M. Steinbach, On PDE solution in transient optimization of gas networks,, J. Comput. Appl. Math., 203 (2007), 345. doi: 10.1016/j.cam.2006.04.018. Google Scholar

[31]

G. Steinebach, S. Rademacher, P. Rentrop and M. Schulz, Mechanisms of coupling in river flow simulation systems,, J. Comput. Appl. Math., 168 (2004), 459. doi: 10.1016/j.cam.2003.12.008. Google Scholar

[32]

D. D. Sworder and V. G. Robinson, Feedback regulators for jump parameter systems with state and control depend transistion rates,, IEEE Trans. Automat. Control, AC-18 (1973), 355. doi: 10.1109/TAC.1973.1100343. Google Scholar

[33]

A. S. Willsky and B. C. Rogers, Stochastic stability research for complex power systems,, DOE Contract, (): 01. Google Scholar

[34]

G. G. Yin and Q. Zhang, "Discrete-Time Markov Chains. Two-Time-Scale Methods and Applications,'', Applications of Mathematics (New York), 55 (2005). Google Scholar

show all references

References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains,, SIAM J. Appl. Math., 66 (2006), 896. doi: 10.1137/040604625. Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Networks and Heterogenous Media, 1 (2006), 41. doi: 10.3934/nhm.2006.1.41. Google Scholar

[3]

M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations,, Networks and Heterogenous Media, 1 (2006), 295. doi: 10.3934/nhm.2006.1.295. Google Scholar

[4]

S. Battiston, D. Delli Gatti, M. Gallegati, B. Greenwald and J. E. Stiglitz, Credit chains and bankruptcy propagation in production networks,, J. Economic Dynamics and Control, 31 (2007), 2061. doi: 10.1016/j.jedc.2007.01.004. Google Scholar

[5]

G. Bretti, C. D'Apice, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics,, Networks and Heterogeneous Media, 2 (2007), 661. doi: 10.3934/nhm.2007.2.661. Google Scholar

[6]

G. Coclite, M. Garavello and B. Piccoli, Traffic flow on road networks,, SIAM J. Mathematical Analysis, 36 (2005), 1862. doi: 10.1137/S0036141004402683. Google Scholar

[7]

C. D'Apice and R. Manzo, A fluid-dynamic model for supply chain,, Networks and Heterogeneous Media, 1 (2006), 379. doi: 10.3934/nhm.2006.1.379. Google Scholar

[8]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models. With discussion,, J. Royal Statistical Society Ser. B, 46 (1984), 353. Google Scholar

[9]

M. H. A. Davis, "Markov Models and Optimisation,", Monograph on Statistics and Applied Probability, 49 (1993). Google Scholar

[10]

P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns,, SIAM J. Appl. Math., 68 (2007), 59. doi: 10.1137/060674302. Google Scholar

[11]

A. Fügenschuh, M. Herty and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks,, SIAM J. Optimization, 16 (2006), 1155. Google Scholar

[12]

C. W. Gardiner, "Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,'' 3rd edition,, Springer Series in Synergetics, 13 (2004). Google Scholar

[13]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions,, J. Computational Phys., 22 (1976), 403. doi: 10.1016/0021-9991(76)90041-3. Google Scholar

[14]

D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems,, J. Chem. Phys., 115 (2001), 1716. doi: 10.1063/1.1378322. Google Scholar

[15]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains,, Comm. Math. Sci., 3 (2005), 545. Google Scholar

[16]

S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks,, Comm. Math. Sci., 4 (2006), 315. Google Scholar

[17]

S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks,, European J. of Operational Research, 202 (2010), 456. doi: 10.1016/j.ejor.2009.05.028. Google Scholar

[18]

M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks,, J. Optimization Theory and Application, 126 (2005), 589. doi: 10.1007/s10957-005-5499-z. Google Scholar

[19]

D. Helbing, "Verkehrsdynamik,'', Springer Verlag, (1997). doi: 10.1007/978-3-642-59063-4. Google Scholar

[20]

D. Helbing, S. Lämmer and T. Seidel, Physics, stability and dynamics of supply chains,, Physical Review E, 70 (2004), 066116. doi: 10.1103/PhysRevE.70.066116. Google Scholar

[21]

M. Herty and A. Klar, Modeling, simulation and optimization of traffic flow networks,, SIAM J. Scientific Computing, 25 (2003), 1066. doi: 10.1137/S106482750241459X. Google Scholar

[22]

M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations,, SIAM J. Mathematical Analysis, 39 (2007), 160. doi: 10.1137/060659478. Google Scholar

[23]

T. Kazangey and D. D. Sworder, Effective federal policies for regulating residential housing,, Proc. Summer Computer Simulation Conf., (1971), 1120. Google Scholar

[24]

F. P. Kelly, S. Zachary and I. Ziedins, eds., "Stochastic Networks: Theory and Apllications,", Oxford University Press, (2002). Google Scholar

[25]

C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal control for continuous supply network models,, Networks and Heterogenous Media, 1 (2006), 675. doi: 10.3934/nhm.2006.1.675. Google Scholar

[26]

G. Leugering and E. Schmidt, On the modelling and stabilization of flows in networks of open channels,, SIAM J. Control and Optimization, 41 (2002), 164. Google Scholar

[27]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,'', Imperial College Press, (2006). Google Scholar

[28]

M. Mariton, "Jump Linear Systems in Automatic Control,'', Marcel Dekker, (1990). Google Scholar

[29]

A. Martin, M. Möller and S. Moritz, Mixed integer models for the stationary case of gas network optimization,, Math. Programming, 105 (2006), 563. doi: 10.1007/s10107-005-0665-5. Google Scholar

[30]

M. Steinbach, On PDE solution in transient optimization of gas networks,, J. Comput. Appl. Math., 203 (2007), 345. doi: 10.1016/j.cam.2006.04.018. Google Scholar

[31]

G. Steinebach, S. Rademacher, P. Rentrop and M. Schulz, Mechanisms of coupling in river flow simulation systems,, J. Comput. Appl. Math., 168 (2004), 459. doi: 10.1016/j.cam.2003.12.008. Google Scholar

[32]

D. D. Sworder and V. G. Robinson, Feedback regulators for jump parameter systems with state and control depend transistion rates,, IEEE Trans. Automat. Control, AC-18 (1973), 355. doi: 10.1109/TAC.1973.1100343. Google Scholar

[33]

A. S. Willsky and B. C. Rogers, Stochastic stability research for complex power systems,, DOE Contract, (): 01. Google Scholar

[34]

G. G. Yin and Q. Zhang, "Discrete-Time Markov Chains. Two-Time-Scale Methods and Applications,'', Applications of Mathematics (New York), 55 (2005). Google Scholar

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