Time-continuous production networks with random breakdowns

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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

Abstract
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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 and 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), 896-920. doi: 10.1137/040604625.
[2]	M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Networks and Heterogenous Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.
[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), 295-314. doi: 10.3934/nhm.2006.1.295.
[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), 2061-2084. doi: 10.1016/j.jedc.2007.01.004.
[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), 661-694. doi: 10.3934/nhm.2007.2.661.
[6]	G. Coclite, M. Garavello and B. Piccoli, Traffic flow on road networks, SIAM J. Mathematical Analysis, 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.
[7]	C. D'Apice and R. Manzo, A fluid-dynamic model for supply chain, Networks and Heterogeneous Media, 1 (2006), 379-398. doi: 10.3934/nhm.2006.1.379.
[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), 353-388.
[9]	M. H. A. Davis, "Markov Models and Optimisation," Monograph on Statistics and Applied Probability, 49, Chapmand & Hall, London, 1993.
[10]	P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Appl. Math., 68 (2007), 59-79. doi: 10.1137/060674302.
[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), 1155-1176.
[12]	C. W. Gardiner, "Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,'' 3^rd edition, Springer Series in Synergetics, 13, Springer-Verlag, Berlin, 2004.
[13]	D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Computational Phys., 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.
[14]	D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115 (2001), 1716-1733. doi: 10.1063/1.1378322.
[15]	S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Comm. Math. Sci., 3 (2005), 545-559.
[16]	S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks, Comm. Math. Sci., 4 (2006), 315-330.
[17]	S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks, European J. of Operational Research, 202 (2010), 456-465. doi: 10.1016/j.ejor.2009.05.028.
[18]	M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optimization Theory and Application, 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.
[19]	D. Helbing, "Verkehrsdynamik,'' Springer Verlag, New York, Berlin, Heidelberg, 1997. doi: 10.1007/978-3-642-59063-4.
[20]	D. Helbing, S. Lämmer and T. Seidel, Physics, stability and dynamics of supply chains, Physical Review E, 70 (2004), 066116-066120. doi: 10.1103/PhysRevE.70.066116.
[21]	M. Herty and A. Klar, Modeling, simulation and optimization of traffic flow networks, SIAM J. Scientific Computing, 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.
[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), 160-173. doi: 10.1137/060659478.
[23]	T. Kazangey and D. D. Sworder, Effective federal policies for regulating residential housing, Proc. Summer Computer Simulation Conf., (1971), 1120-1128.
[24]	F. P. Kelly, S. Zachary and I. Ziedins, eds., "Stochastic Networks: Theory and Apllications," Oxford University Press, 2002.
[25]	C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal control for continuous supply network models, Networks and Heterogenous Media, 1 (2006), 675-688. doi: 10.3934/nhm.2006.1.675.
[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), 164-180.
[27]	X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,'' Imperial College Press, London, 2006.
[28]	M. Mariton, "Jump Linear Systems in Automatic Control,'' Marcel Dekker, 1990.
[29]	A. Martin, M. Möller and S. Moritz, Mixed integer models for the stationary case of gas network optimization, Math. Programming, 105 (2006), 563-582. doi: 10.1007/s10107-005-0665-5.
[30]	M. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361. doi: 10.1016/j.cam.2006.04.018.
[31]	G. Steinebach, S. Rademacher, P. Rentrop and M. Schulz, Mechanisms of coupling in river flow simulation systems, J. Comput. Appl. Math., 168 (2004), 459-470. doi: 10.1016/j.cam.2003.12.008.
[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), 355-360. doi: 10.1109/TAC.1973.1100343.
[33]	A. S. Willsky and B. C. Rogers, Stochastic stability research for complex power systems, DOE Contract, LIDS, MIT, Rep., ET-76-C-01-2295.
[34]	G. G. Yin and Q. Zhang, "Discrete-Time Markov Chains. Two-Time-Scale Methods and Applications,'' Applications of Mathematics (New York), 55, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2005.

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), 896-920. doi: 10.1137/040604625.
[2]	M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Networks and Heterogenous Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.
[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), 295-314. doi: 10.3934/nhm.2006.1.295.
[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), 2061-2084. doi: 10.1016/j.jedc.2007.01.004.
[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), 661-694. doi: 10.3934/nhm.2007.2.661.
[6]	G. Coclite, M. Garavello and B. Piccoli, Traffic flow on road networks, SIAM J. Mathematical Analysis, 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.
[7]	C. D'Apice and R. Manzo, A fluid-dynamic model for supply chain, Networks and Heterogeneous Media, 1 (2006), 379-398. doi: 10.3934/nhm.2006.1.379.
[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), 353-388.
[9]	M. H. A. Davis, "Markov Models and Optimisation," Monograph on Statistics and Applied Probability, 49, Chapmand & Hall, London, 1993.
[10]	P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Appl. Math., 68 (2007), 59-79. doi: 10.1137/060674302.
[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), 1155-1176.
[12]	C. W. Gardiner, "Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,'' 3^rd edition, Springer Series in Synergetics, 13, Springer-Verlag, Berlin, 2004.
[13]	D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Computational Phys., 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.
[14]	D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115 (2001), 1716-1733. doi: 10.1063/1.1378322.
[15]	S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Comm. Math. Sci., 3 (2005), 545-559.
[16]	S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks, Comm. Math. Sci., 4 (2006), 315-330.
[17]	S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks, European J. of Operational Research, 202 (2010), 456-465. doi: 10.1016/j.ejor.2009.05.028.
[18]	M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optimization Theory and Application, 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.
[19]	D. Helbing, "Verkehrsdynamik,'' Springer Verlag, New York, Berlin, Heidelberg, 1997. doi: 10.1007/978-3-642-59063-4.
[20]	D. Helbing, S. Lämmer and T. Seidel, Physics, stability and dynamics of supply chains, Physical Review E, 70 (2004), 066116-066120. doi: 10.1103/PhysRevE.70.066116.
[21]	M. Herty and A. Klar, Modeling, simulation and optimization of traffic flow networks, SIAM J. Scientific Computing, 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.
[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), 160-173. doi: 10.1137/060659478.
[23]	T. Kazangey and D. D. Sworder, Effective federal policies for regulating residential housing, Proc. Summer Computer Simulation Conf., (1971), 1120-1128.
[24]	F. P. Kelly, S. Zachary and I. Ziedins, eds., "Stochastic Networks: Theory and Apllications," Oxford University Press, 2002.
[25]	C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal control for continuous supply network models, Networks and Heterogenous Media, 1 (2006), 675-688. doi: 10.3934/nhm.2006.1.675.
[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), 164-180.
[27]	X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,'' Imperial College Press, London, 2006.
[28]	M. Mariton, "Jump Linear Systems in Automatic Control,'' Marcel Dekker, 1990.
[29]	A. Martin, M. Möller and S. Moritz, Mixed integer models for the stationary case of gas network optimization, Math. Programming, 105 (2006), 563-582. doi: 10.1007/s10107-005-0665-5.
[30]	M. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361. doi: 10.1016/j.cam.2006.04.018.
[31]	G. Steinebach, S. Rademacher, P. Rentrop and M. Schulz, Mechanisms of coupling in river flow simulation systems, J. Comput. Appl. Math., 168 (2004), 459-470. doi: 10.1016/j.cam.2003.12.008.
[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), 355-360. doi: 10.1109/TAC.1973.1100343.
[33]	A. S. Willsky and B. C. Rogers, Stochastic stability research for complex power systems, DOE Contract, LIDS, MIT, Rep., ET-76-C-01-2295.
[34]	G. G. Yin and Q. Zhang, "Discrete-Time Markov Chains. Two-Time-Scale Methods and Applications,'' Applications of Mathematics (New York), 55, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2005.

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