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Shock formation in a traffic flow model with Arrhenius look-ahead dynamics
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 |
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,'' 3rd 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,'' 3rd 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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