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Semi-Markovian capacities in production network models
University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany |
In this paper, we focus on production network models based on ordinary and partial differential equations that are coupled to semi-Markovian failure rates for the processor capacities. This modeling approach allows for intermediate capacity states in the range of total breakdown to full capacity, where operating and down times might be arbitrarily distributed. The mathematical challenge is to combine the theory of semi-Markovian processes within the framework of conservation laws. We show the existence and uniqueness of such stochastic network solutions, present a suitable simulation method and explain the link to the common queueing theory. A variety of numerical examples emphasizes the characteristics of the proposed approach.
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] |
J. Banks, J. S. Carson, B. L. Nelson and D. M. Nicol, Discrete-Event System Simulation 5th edition, Pearson, 2010. |
[3] |
C. D'Apice, S. Göttlich, M. Herty and B. Piccoli,
Modeling, Simulation, and Optimization of Supply Chains 1st edition, SIAM, Philadelphia, 2010.
doi: 10.1137/1.9780898717600. |
[4] |
C. D'Apice, P. I. Kogut and R. Manzo,
On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains, Netw. Heterog. Media, 9 (2014), 501-518.
doi: 10.3934/nhm.2014.9.501. |
[5] |
C. D'Apice, R. Manzo and B. Piccoli,
Modelling supply networks with partial differential equations, Quart. Appl. Math., 67 (2009), 419-440.
doi: 10.1090/S0033-569X-09-01129-1. |
[6] |
C. D'Apice, R. Manzo and B. Piccoli,
Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks, J. Math. Anal. Appl., 362 (2010), 374-386.
doi: 10.1016/j.jmaa.2009.07.058. |
[7] |
C. D'Apice, R. Manzo and B. Piccoli,
Optimal input flows for a PDE-ODE model of supply chains, Commun. Math. Sci., 10 (2012), 1225-1240.
doi: 10.4310/CMS.2012.v10.n4.a10. |
[8] |
C. D'Apice, R. Manzo and B. Piccoli,
Numerical schemes for the optimal input flow of a supply chain, SIAM J. Numer. Anal., 51 (2013), 2634-2650.
doi: 10.1137/120889721. |
[9] |
M. H. A. Davis,
Piecewise-deterministic {M}arkov processes: A general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388.
|
[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] |
L. Forestier-Coste, S. Göttlich and M. Herty,
Data-fitted second-order macroscopic production models, SIAM J. Appl. Math., 75 (2015), 999-1014.
doi: 10.1137/140989832. |
[12] |
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. |
[13] |
D. T. Gillespie,
Approximate accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115 (2001), 1716-1733.
doi: 10.1063/1.1378322. |
[14] |
S. Göttlich, M. Herty and A. Klar,
Network models for supply chains, Commun. Math. Sci., 3 (2005), 545-559.
doi: 10.4310/CMS.2005.v3.n4.a5. |
[15] |
S. Göttlich, M. Herty and C. Ringhofer,
Optimization of order policies in supply networks, European J. Oper. Res., 202 (2010), 456-465.
doi: 10.1016/j.ejor.2009.05.028. |
[16] |
S. Göttlich, A. Klar and S. Tiwari,
Complex material flow problems: A multi-scale model hierarchy and particle methods, J. Engrg. Math, 92 (2015), 15-29.
doi: 10.1007/s10665-014-9767-5. |
[17] |
S. Göttlich, S. Martin and T. Sickenberger,
Time-continuous production networks with random breakdowns, Netw. Heterog. Media, 6 (2011), 695-714.
doi: 10.3934/nhm.2011.6.695. |
[18] |
F. Grabski,
Semi-Markov failure rates processes, Appl. Math. Comput., 217 (2011), 9956-9965.
doi: 10.1016/j.amc.2011.04.055. |
[19] |
F. Grabski, Semi-Markov Processes: Applications in System Reliability and Maintenance, 1st edition, Elsevier, Amsterdam, 2015.
doi: 10.1007/978-1-4612-0873-0. |
[20] |
D. Gross, J. F. Shortle, J. M. Thompson and C. M. Harris, Fundamentals of Queueing Theory, 4th edition, John Wiley & Sons, Inc. , Hoboken, NJ, 2008.
doi: 10.1002/9781118625651. |
[21] |
B. Harlamov, Continuous Semi-{M}arkov Processes, 1st edition, ISTE, London; John Wiley & Sons, Inc. , Hoboken, NJ, 2008.
doi: 10.1002/9780470610923. |
[22] |
M. Kolonko, Stochastische Simulation, (German) [Stochastic Simulation], 1st edition, Vieweg+Teubner Verlag, Wiesbaden, 2008.
doi: 10.1007/978-3-8348-9290-4. |
[23] |
A. M. Lee, Applied Queueing Theory, Reprint edition, Macmillan [u. a. ], London [u. a. ], 1966.
doi: 10.1007/978-1-349-00273-3. |
[24] |
L. Lipsky, Queueing Theory, 2nd edition, Springer, New York, 2009.
doi: 10.1007/978-0-387-49706-8. |
[25] |
J. Medhi,
Stochastic Processes, 3rd edition, New Age Science, Tunbridge Wells, 2010. |
[26] |
J. R. Norris, Markov Chains, Reprint edition, Cambridge University Press, Cambridge, 1998. |
[27] |
R. Pyke,
Markov renewal processes: Definitions and preliminary properties, Ann. Math. Statist., 32 (1961), 1231-1242.
doi: 10.1214/aoms/1177704863. |
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] |
J. Banks, J. S. Carson, B. L. Nelson and D. M. Nicol, Discrete-Event System Simulation 5th edition, Pearson, 2010. |
[3] |
C. D'Apice, S. Göttlich, M. Herty and B. Piccoli,
Modeling, Simulation, and Optimization of Supply Chains 1st edition, SIAM, Philadelphia, 2010.
doi: 10.1137/1.9780898717600. |
[4] |
C. D'Apice, P. I. Kogut and R. Manzo,
On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains, Netw. Heterog. Media, 9 (2014), 501-518.
doi: 10.3934/nhm.2014.9.501. |
[5] |
C. D'Apice, R. Manzo and B. Piccoli,
Modelling supply networks with partial differential equations, Quart. Appl. Math., 67 (2009), 419-440.
doi: 10.1090/S0033-569X-09-01129-1. |
[6] |
C. D'Apice, R. Manzo and B. Piccoli,
Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks, J. Math. Anal. Appl., 362 (2010), 374-386.
doi: 10.1016/j.jmaa.2009.07.058. |
[7] |
C. D'Apice, R. Manzo and B. Piccoli,
Optimal input flows for a PDE-ODE model of supply chains, Commun. Math. Sci., 10 (2012), 1225-1240.
doi: 10.4310/CMS.2012.v10.n4.a10. |
[8] |
C. D'Apice, R. Manzo and B. Piccoli,
Numerical schemes for the optimal input flow of a supply chain, SIAM J. Numer. Anal., 51 (2013), 2634-2650.
doi: 10.1137/120889721. |
[9] |
M. H. A. Davis,
Piecewise-deterministic {M}arkov processes: A general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388.
|
[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] |
L. Forestier-Coste, S. Göttlich and M. Herty,
Data-fitted second-order macroscopic production models, SIAM J. Appl. Math., 75 (2015), 999-1014.
doi: 10.1137/140989832. |
[12] |
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. |
[13] |
D. T. Gillespie,
Approximate accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115 (2001), 1716-1733.
doi: 10.1063/1.1378322. |
[14] |
S. Göttlich, M. Herty and A. Klar,
Network models for supply chains, Commun. Math. Sci., 3 (2005), 545-559.
doi: 10.4310/CMS.2005.v3.n4.a5. |
[15] |
S. Göttlich, M. Herty and C. Ringhofer,
Optimization of order policies in supply networks, European J. Oper. Res., 202 (2010), 456-465.
doi: 10.1016/j.ejor.2009.05.028. |
[16] |
S. Göttlich, A. Klar and S. Tiwari,
Complex material flow problems: A multi-scale model hierarchy and particle methods, J. Engrg. Math, 92 (2015), 15-29.
doi: 10.1007/s10665-014-9767-5. |
[17] |
S. Göttlich, S. Martin and T. Sickenberger,
Time-continuous production networks with random breakdowns, Netw. Heterog. Media, 6 (2011), 695-714.
doi: 10.3934/nhm.2011.6.695. |
[18] |
F. Grabski,
Semi-Markov failure rates processes, Appl. Math. Comput., 217 (2011), 9956-9965.
doi: 10.1016/j.amc.2011.04.055. |
[19] |
F. Grabski, Semi-Markov Processes: Applications in System Reliability and Maintenance, 1st edition, Elsevier, Amsterdam, 2015.
doi: 10.1007/978-1-4612-0873-0. |
[20] |
D. Gross, J. F. Shortle, J. M. Thompson and C. M. Harris, Fundamentals of Queueing Theory, 4th edition, John Wiley & Sons, Inc. , Hoboken, NJ, 2008.
doi: 10.1002/9781118625651. |
[21] |
B. Harlamov, Continuous Semi-{M}arkov Processes, 1st edition, ISTE, London; John Wiley & Sons, Inc. , Hoboken, NJ, 2008.
doi: 10.1002/9780470610923. |
[22] |
M. Kolonko, Stochastische Simulation, (German) [Stochastic Simulation], 1st edition, Vieweg+Teubner Verlag, Wiesbaden, 2008.
doi: 10.1007/978-3-8348-9290-4. |
[23] |
A. M. Lee, Applied Queueing Theory, Reprint edition, Macmillan [u. a. ], London [u. a. ], 1966.
doi: 10.1007/978-1-349-00273-3. |
[24] |
L. Lipsky, Queueing Theory, 2nd edition, Springer, New York, 2009.
doi: 10.1007/978-0-387-49706-8. |
[25] |
J. Medhi,
Stochastic Processes, 3rd edition, New Age Science, Tunbridge Wells, 2010. |
[26] |
J. R. Norris, Markov Chains, Reprint edition, Cambridge University Press, Cambridge, 1998. |
[27] |
R. Pyke,
Markov renewal processes: Definitions and preliminary properties, Ann. Math. Statist., 32 (1961), 1231-1242.
doi: 10.1214/aoms/1177704863. |






















Case 1 | Case 2 | |
2.2357 | 2.2197 | |
0.0529 | 0.0466 |
Case 1 | Case 2 | |
2.2357 | 2.2197 | |
0.0529 | 0.0466 |
Processor e | 1 | 2 | 3 | 4 | 5 |
MTBF | 0.95 | | 0.85 | 1.9 | 0.95 |
MRT | 0.05 | 0 | 0.15 | 0.10 | 0.05 |
Processor e | 1 | 2 | 3 | 4 | 5 |
MTBF | 0.95 | | 0.85 | 1.9 | 0.95 |
MRT | 0.05 | 0 | 0.15 | 0.10 | 0.05 |
| |||
1.3361 | 1.7391 | 2.8382 | |
0.3278 | 1.2588 | 5.1694 |
| |||
1.3361 | 1.7391 | 2.8382 | |
0.3278 | 1.2588 | 5.1694 |
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