Formal relation of Markov renewal theory and supplementary variables in the analysis of stochastic Petri nets

Non-Markovian stochastic Petri nets have been investigated mainly by means of Markov renewal theory and by the method of supplementary variables. Both approaches provide different analytic descriptions of the same system. Numerical algorithms based on these descriptions lead to similar results. Parallel research effort resulted from the fact that an exact relationship of the two was not known. In this paper such a formal relationship is established for Markov regenerative stochastic Petri nets with general preemption policies in both the transient and stationary case. As a by-product, a closed form solution in Laplace domain is derived, which is easier to apply than previously known ones. An example from communications is used for illustrations.