Closed form approximations for steady state probabilities of a controlled fork-join network
Date
2010
Authors
Billington, J.
Gallasch, G.E.
Editors
Dong, J.S.
Zhu, H.
Zhu, H.
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Book chapter
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Source details - Title: Formal Methods and Software Engineering, 2010 / Dong, J.S., Zhu, H. (ed./s), Ch.28, pp.420-435
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Abstract
Our work is motivated by just-in-time manufacturing systems, where goods are produced on demand. We consider a class of products made from two components each manufactured by its own production line. The components are then assembled, requiring synchronisation of the two lines. The production lines are coordinated to ensure that one line does not get ahead of the other by more than a certain number of components, N, a parameter of the system. We assume that the statistics of the processes follow exponential distributions, with requests to manufacture the product arriving at a rate λ 0 and the two production lines having rates λ 1 and λ 2. Generalised Stochastic Petri Nets (GSPN) are used to model this system where N is the initial marking of a control place. TimeNET is used to calculate the stationary token distribution of the GSPN as N increases, revealing convergence of the steady state probabilities. We characterise the range of rates for which useful convergence occurs using a large number of TimeNET runs and show how these results can be used to approximate the steady state probabilities for arbitrarily large N, to a desired level of accuracy. Further, for λ 0 > min(λ 1,λ 2) we discover geometric progressions in the steady state probabilities once they have converged. We use these progressions to derive closed form approximations, the accuracies of which increase as N increases.
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Copyright 2010 Springer-Verlag Berlin Heidelberg