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Stochastic Petri Box Calculus with Discrete Time

In the last decades, a number of stochastic enrichments of process algebras was constructed to allow one for specification of stochastic processes within the well-developed framework of algebraic calculi. In [40], a continuous time stochastic extension of finite Petri box calculus (PBC) was proposed called sPBC. The algebra sPBC has interleaving semantics due to the properties of continuous time distributions. At the same time, PBC has step semantics, and it could be natural to propose its concurrent stochastic enrichment. We construct a discrete time stochastic extension dtsPBC of finite PBC. A step operational semantics is defined in terms of labeled transition systems based on action and inaction rules. A denotational semantics is defined in terms of a subclass of labeled discrete time stochastic Petri nets (LDTSPNs) called discrete time stochastic Petri boxes (dts-boxes). A consistency of both semantics is demonstrated. In addition, we define a variety of probabilistic equivalences that allow one to identify stochastic processes with similar behaviour which are differentiated by too strict notion of the semantic equivalence. The interrelations of all the introduced equivalences are investigated.