Semi-Markovian capacities in production network models

Simone Göttlich; Stephan Knapp

doi:10.3934/dcdsb.2017090

Article Contents

2017, Volume 22, Issue 9: 3235-3258. Doi: 10.3934/dcdsb.2017090

This issue Previous Article Next Article Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones

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
* Corresponding author: Simone Göttlich

Received: June 30, 2016

Revised: November 30, 2016

Published: October 2017

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

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Abstract

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.

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

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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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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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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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