A study of the convergence of steady state probabilities in a closed fork-join network

  • Authors:

  • Guy Edward Gallasch;Jonathan Billington

  • Affiliations:

  • Computer Systems Engineering Centre, School of Electrical and Information Engineering, University of South Australia, SA, Australia;Computer Systems Engineering Centre, School of Electrical and Information Engineering, University of South Australia, SA, Australia

  • Venue:
  • ATVA'10 Proceedings of the 8th international conference on Automated technology for verification and analysis
  • Year:
  • 2010

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Abstract

Fork-join structures are important for modelling parallel and distributed systems where coordination and synchronisation occur, but their performance analysis is difficult. The study presented in this paper is motivated by the need to calculate performance measures for computer controlled (agile) manufacturing systems. We consider the manufacture of a class of products made from two components that are created by two parallel production lines. The components are then assembled, requiring synchronisation of the two lines. The products are only created on request from the client. The production lines need to be controlled so that one line does not get ahead of the other by more than a certain amount, N, a parameter of the system. We model this system with a Generalised Stochastic Petri Net, where N is the initial marking of a control place. The customer requests and the two production line rates are modelled by exponential distributions associated with three timed transitions in the model. We use TimeNET to calculate the stationary token distribution of the GSPN for a wide range of the rates as N increases. This reveals that the steady state probabilities converge. We characterise the range of transition rates for which useful convergence occurs and provide a method for obtaining the steady state probabilities to the desired accuracy for arbitrary N.