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

Time-continuous production networks with random breakdowns

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].
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.

[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.

[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.

[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.

[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.

[6]

G. Coclite, M. Garavello and B. Piccoli, Traffic flow on road networks,, SIAM J. Mathematical Analysis, 36 (2005), 1862. 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. 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.

[9]

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

[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.

[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.

[12]

C. W. Gardiner, "Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,'' 3rd edition,, Springer Series in Synergetics, 13 (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. 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. doi: 10.1063/1.1378322.

[15]

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

[16]

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

[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.

[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.

[19]

D. Helbing, "Verkehrsdynamik,'', Springer Verlag, (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. 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. 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. 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.

[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. 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.

[27]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,'', Imperial College Press, (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. 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. 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. 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. doi: 10.1109/TAC.1973.1100343.

[33]

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

[34]

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

[9]

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

[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.

[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.

[12]

C. W. Gardiner, "Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,'' 3rd edition,, Springer Series in Synergetics, 13 (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. 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. doi: 10.1063/1.1378322.

[15]

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

[16]

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

[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.

[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.

[19]

D. Helbing, "Verkehrsdynamik,'', Springer Verlag, (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. 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. 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. 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.

[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. 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.

[27]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,'', Imperial College Press, (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. 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. 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. 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. doi: 10.1109/TAC.1973.1100343.

[33]

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

[34]

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

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