Time-continuous production networks with random breakdowns

Simone Göttlich; Stephan Martin; Thorsten Sickenberger

doi:10.3934/nhm.2011.6.695

Article Contents

2011, Volume 6, Issue 4: 695-714. Doi: 10.3934/nhm.2011.6.695

This issue Previous Article Shock formation in a traffic flow model with Arrhenius look-ahead dynamics Next Article Ginzburg-Landau model with small pinning domains

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
3.
Maxwell Institute and Heriot-Watt University, Dept. of Mathematics, Edinburgh, EH14 4AS, Scotland

Received: March 31, 2011

Revised: September 30, 2011

Published: November 2011

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Abstract

Our main objective is the modelling and simulation of complex production networks originally introduced in [15, 16] with random breakdowns of individual processors. Similar to [10], the breakdowns of processors are exponentially distributed. The resulting network model consists of coupled system of partial and ordinary differential equations with Markovian switching and its solution is a stochastic process. We show our model to fit into the framework of piecewise deterministic processes, which allows for a deterministic interpretation of dynamics between a multivariate two-state process. We develop an efficient algorithm with an emphasis on accurately tracing stochastic events. Numerical results are presented for three exemplary networks, including a comparison with the long-chain model proposed in [10].

Keywords:

Production networks,
conservation laws,
random breakdowns,
coupled PDE-ODE systems,
Markovian switching,
piecewise deterministic processes (PDPs).

Mathematics Subject Classification: Primary: 90B15, 65C05; Secondary: 65Mxx.

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