Semi-Markovian capacities in production network models

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November 2017, 22(9): 3235-3258. doi: 10.3934/dcdsb.2017090

Semi-Markovian capacities in production network models

Simone Göttlich ^, and Stephan Knapp

University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany

* Corresponding author: Simone Göttlich

Received July 2016 Revised December 2016 Published March 2017

Fund Project: This work was financially supported by the DAAD project "DAAD-PPP VR China" (Projekt-ID: 57215936)

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.

Keywords: Production networks, conservation laws, semi-Markovian processes.

Mathematics Subject Classification: Primary:90B15, 65Cxx; Secondary:65Mxx.

Citation: Simone Göttlich, Stephan Knapp. Semi-Markovian capacities in production network models. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3235-3258. doi: 10.3934/dcdsb.2017090

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 5^th edition, Pearson, 2010.
[3]	C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains 1^st 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, 1^st 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, 4^th edition, John Wiley & Sons, Inc. , Hoboken, NJ, 2008. doi: 10.1002/9781118625651.
[21]	B. Harlamov, Continuous Semi-{M}arkov Processes, 1^st edition, ISTE, London; John Wiley & Sons, Inc. , Hoboken, NJ, 2008. doi: 10.1002/9780470610923.
[22]	M. Kolonko, Stochastische Simulation, (German) [Stochastic Simulation], 1^st 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, 2^nd edition, Springer, New York, 2009. doi: 10.1007/978-0-387-49706-8.
[25]	J. Medhi, Stochastic Processes, 3^rd 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 5^th edition, Pearson, 2010.
[3]	C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains 1^st 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, 1^st 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, 4^th edition, John Wiley & Sons, Inc. , Hoboken, NJ, 2008. doi: 10.1002/9781118625651.
[21]	B. Harlamov, Continuous Semi-{M}arkov Processes, 1^st edition, ISTE, London; John Wiley & Sons, Inc. , Hoboken, NJ, 2008. doi: 10.1002/9780470610923.
[22]	M. Kolonko, Stochastische Simulation, (German) [Stochastic Simulation], 1^st 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, 2^nd edition, Springer, New York, 2009. doi: 10.1007/978-0-387-49706-8.
[25]	J. Medhi, Stochastic Processes, 3^rd 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.

Figure 1. General idea of a semi-Markov process

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Figure 2. Sample path and its pseudo inverse

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Figure 3. Comparison with $\lambda = 0.1$

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Figure 4. Comparison with $\lambda = 0.5$

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Figure 5. Comparison with $\lambda = 0.75$

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Figure 6. Comparison with $\lambda = 1.25$

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Figure 7. Graph representation of the CTMCs

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Figure 8. Difference of the sampled mean an variance densities

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Figure 9. Sampled mean and variance of the queue-loads

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Figure 10. Sample mean and variance of the network outflow

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Figure 11. Difference of the sample mean and variance densities

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Figure 12. Sample mean and variance of the queue-loads

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Figure 13. Sample mean and variance of the network outflow

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Figure 14. Histogram of production times

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Figure 15. Different pdf of the gamma distribution

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Figure 16. erial network with five processors

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Figure 17. Sample mean of the density in the exponential case

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Figure 18. Sampled mean of the density with $\alpha = 4$ and $\alpha = 0.25$

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Figure 19. Sampled mean and variance of the network outflow

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Figure 20. Comparison of the sample mean and variance of the first processor capacity process

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Figure 21. Comparison of the sample mean of the network outflow and the network queue-loads

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Figure 22. Comparison of the sample variance of the network outflow and the network queue-loads

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Figure 23. Comparison of the production time for 0.9 amount of goods

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Table 1. Sample mean and variance of the production time

	Case 1	Case 2
$\overline{\tau}_{prod}(\infty)$	2.2357	2.2197
$\sigma^2(\tau_{prod}(\infty))$	0.0529	0.0466

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	Case 1	Case 2
$\overline{\tau}_{prod}(\infty)$	2.2357	2.2197
$\sigma^2(\tau_{prod}(\infty))$	0.0529	0.0466

Table 2. Parameters of the network model with five processors

Processor e	1	2	3	4	5
MTBF	0.95	$\infty$	0.85	1.9	0.95
MRT	0.05	0	0.15	0.10	0.05

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Processor e	1	2	3	4	5
MTBF	0.95	$\infty$	0.85	1.9	0.95
MRT	0.05	0	0.15	0.10	0.05

Table 3. Sample mean and variance of the network queue-load

	$\alpha = \frac{1}{4}$	$\alpha = 1$	$\alpha = 4$
$\overline{q^{net}(4)}$	1.3361	1.7391	2.8382
$\sigma^2(q^{net}(4))$	0.3278	1.2588	5.1694

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	$\alpha = \frac{1}{4}$	$\alpha = 1$	$\alpha = 4$
$\overline{q^{net}(4)}$	1.3361	1.7391	2.8382
$\sigma^2(q^{net}(4))$	0.3278	1.2588	5.1694

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