# Discrete time stochastic and deterministic Petri box calculus

## TL;DR

This paper introduces an extended Petri calculus with deterministic timing for multiactions, combining stochastic and deterministic models, and analyzes their semantics and performance through probabilistic and Markov chain methods.

## Contribution

It extends discrete time stochastic Petri calculus with deterministic timing, providing new semantics and performance analysis tools.

## Key findings

- Semantics are consistent between operational and denotational models.
- Performance evaluation is achieved through semi-Markov and Markov chain analysis.
- The extended calculus captures both stochastic and deterministic timing behaviors.

## Abstract

We propose an extension with deterministically timed multiactions of discrete time stochastic and immediate Petri box calculus (dtsiPBC), previously presented by I.V. Tarasyuk, H. Maci\`a and V. Valero. In dtsdPBC, non-negative integers specify multiactions with fixed (including zero) time delays. The step operational semantics is constructed via labeled probabilistic transition systems. The denotational semantics is defined on the basis of a subclass of labeled discrete time stochastic Petri nets with deterministic transitions. The consistency of both semantics is demonstrated. In order to evaluate performance, the corresponding semi-Markov chains and (reduced) discrete time Markov chains are analyzed.