Phase-type approximation of stochastic Petri nets for analysis of manufacturing systems

NonMarkovian stochastic Petri nets (SPN) have received special attention due to their functionality in reflecting nonexponential dynamic behavior encountered in modeling and analysis of real systems. In this paper, a novel analysis approach, based on phase-type approximation, is proposed to provide transient and steady-state probabilities and determine performance measures of these nonMarkovian SPN. The approach can accommodate a wide variety of nonexponential distributions and provide a stronger mechanism than other methods proposed to date for analyzing system performance. The proposed procedure primarily consists of three steps. First, all generally distributed transitions are fitted with phase-type transitions. Next, the nonMarkovian SPN with the approximated phase-type transitions is converted into a Markov chain. Last, transient-state probabilities are obtained by employing the uniformization method and steady-state probabilities are determined by utilizing the preconditioned biconjugate gradient method. Pertinent performance measures can be computed by using these probabilities. The proposed methodology is validated through a real example with respect to its accuracy and speed.