Kronecker representation of stochastic Petri nets with discrete PH distributions

In recent years, several classes of Stochastic Petri Net (SPN) models have been elaborated which incorporate some non-exponential characteristics in their definition. Among the various approaches that have been proposed in the literature for handling non-exponential SPNs, the paper investigates the class of models in which the firing time assigned to each transition is a Discrete Phase Type (DPH) random variable, so that the evolution of the marking process is mapped into an expanded discrete-time Markov chain (DTMC). In order to alleviate the state space explosion problem, the expanded state space is expressed via Kronecker algebra operators, starting from the knowledge of the reachability graph of the untimed PN and the DPH random firing times assigned to each PN-transition. The discrete case is very appealing since it allows to mix distributions with finite and infinite support. However the problem of simultaneous firings arises, and the related semantics must be carefully considered.