Semi-Markovian capacities in production network models

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.