Phase-type approximation of stochastic Petri nets for analysis of manufacturing systems
Shang-Tae Yee, José A. Ventura · IEEE Transactions on Robotics and Automation · 2000
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.