# American Institute of Mathematical Sciences

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

## Semi-Markovian capacities in production network models

 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.

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:

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##### References:
General idea of a semi-Markov process
Sample path and its pseudo inverse
Comparison with $\lambda = 0.1$
Comparison with $\lambda = 0.5$
Comparison with $\lambda = 0.75$
Comparison with $\lambda = 1.25$
Graph representation of the CTMCs
Difference of the sampled mean an variance densities
Sampled mean and variance of the queue-loads
Sample mean and variance of the network outflow
Difference of the sample mean and variance densities
Sample mean and variance of the queue-loads
Sample mean and variance of the network outflow
Histogram of production times
Different pdf of the gamma distribution
erial network with five processors
Sample mean of the density in the exponential case
Sampled mean of the density with $\alpha = 4$ and $\alpha = 0.25$
Sampled mean and variance of the network outflow
Comparison of the sample mean and variance of the first processor capacity process
Comparison of the sample mean of the network outflow and the network queue-loads
Comparison of the sample variance of the network outflow and the network queue-loads
Comparison of the production time for 0.9 amount of goods
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
 Case 1 Case 2 $\overline{\tau}_{prod}(\infty)$ 2.2357 2.2197 $\sigma^2(\tau_{prod}(\infty))$ 0.0529 0.0466
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 0.1 0.05
 Processor e 1 2 3 4 5 MTBF 0.95 $\infty$ 0.85 1.9 0.95 MRT 0.05 0 0.15 0.1 0.05
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
 $\alpha = \frac{1}{4}$ $\alpha = 1$ $\alpha = 4$ $\overline{q^{net}(4)}$ 1.3361 1.7391 2.8382 $\sigma^2(q^{net}(4))$ 0.3278 1.2588 5.1694
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