Fluid stochastic event graphs for optimisation of failure-prone manufacturing systems

This paper addresses the performance evaluation and optimisation of failure-prone manufacturing systems. For this purpose, we propose a fluid stochastic event graph model that is a decision-free Petri net and can be used to model transfer lines, kanban systems, etc. In this model, tokens are considered as continuous flows. A transition can be in operating state or in failure state. A transition in operating state can fire at its maximal speed and a transition in failure state cannot fire. Transitions between failure and operating states are independent of the firing conditions and the sojourn time in each state is a random variable of general distribution. The system is hybrid in the sense that it has both continuous dynamic as a result of continuous material flow and discrete events including failures and repairs. For the purpose of performance evaluation, a set of evolution equations that determines continuous state variables at epochs of discrete events is established. Based on the evolution equations, we prove that the cumulative firing of transitions are Lipschitz continuous, non-decreasing and concave functions of some system parameters including maximal firing rates and the initial marking. Gradient estimators of the cumulative firings with respect to the system parameters are derived. Ergodic properties of the stochastic process are established. Unbiasedness and strong consistency of the gradient estimators are proved. Finally, an optimisation problem of the initial marking that maximises a concave function of throughput rate and the initial marking is addressed.