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Stochastic Petri nets (SPN) or Markov models (MC) are often more effective in the reliability analysis of discrete event systems. However, they present a problem of combinatorial explosion of the number of states when the systems present several interdependent components. This problem limits the use of the MC. The SPN is considered a Markov estimator, but it has a slow convergence in the calculations of stationary state probabilities. The continuous Petri nets (CPN) are developed to accelerate this convergence by capturing the SPN behaviour. This study considers three different Petri nets (PN) types. Simulations with the MC, SPN, and CPN, are presented and compared in different PN types. The obtained results show that the MC and the SPN have identical behaviour in the general case. Furthermore, the CPN exhibits identical behaviour in the first type, which is not the case in the other two types.

The fluidification of stochastic discrete event models is a more interesting method to overcome the problem of combinatorial explosion of states. It allows to relax the conditions of stochastic Petri nets to continuous Petri nets, in order to accelerate the slow convergence of stochastic simulations. In our study, we will study an example of a large manufacturing workshop, in order to give limitations of the direct fluidification which does not always lead to the same behaviour between the two models. Secondly, we propose a numerical approach called adaptive that devotes to the adaptation of the maximum speeds of crossing the transitions that is considered as functions depending on the time. Consequently, this approach shows an excellent convergence of the continuous model to the stochastic one whatever the type of network and whatever the initial marking.

Salinity is one of the major factors that limit tomato growth and productivity, causing morphological and physiological changes and impacting plant metabolism. Seaweed extracts can reduce these harmful effects. Therefore, in this study we examined the effect of Cystoseira tamariscifoliaextract (CTE) on NaCl stress tolerance in tomato plants. We assessed the effect of different concentrations of CTE on germination parameters of tomato seeds to determine the most effective concentrations. Then, we applied the chosen concentrations (2%, 5%, and 10%) of CTE as soil drench to tomato seedlings (Solanum lycopersicum L.) grown under salinity. Our findings revealed a decrease in the growth of tomato plants exposed to 50 mM of NaCl compared to unstressed ones. However, CTE supplementation, especially at 2%, to the stressed plants increased the plant height by 32.24% compared to stressed control without treatment and elevated biomass and chlorophyll content. Additionally, CTE decreased hydrogen peroxide and malondialdehyde accumulation and increased the activities of antioxidant enzymes: superoxide dismutase (SOD), glutathione peroxidase (GPx), and glutathione S-transferase (GST). Moreover, CTE supplementation regulated the alterations in carbon and nitrogen metabolism by increasing the activity of carbon–nitrogen enzymes: phosphoenolpyruvate carboxylase (PEPC), malate dehydrogenase (NAD-MDH), glutamine synthase (GS), glutamate dehydrogenase (GDH), and aspartate aminotransferase (AAT). Furthermore, CTE application increased notably the content of indole acetic acid, soluble sugars, and amino acids and improved the expression of antioxidant metabolites like flavonoids and polyphenols. Overall, our investigations demonstrate that CTE can be used as biostimulant to enhance the salt stress tolerance of tomato plants.

In our study, we were interested in the reliability of large discrete systems. These studies can be based on Markov models or on Stochastic Petri Nets (SPNs) that are generally used for the analysis and synthesis of the models used in the different phases of a system's life. Markov models or SPNs are perfect for many cases, still, they suffer from the combinatorial explosion when analytically their state numbers increase as the complexity of the dynamic systems grows accordingly with their components. Such issue reflects itself in the slowness of these models to accomplish convergence. These different modelling tools make it possible to deduce the average behaviour and to obtain the performance indicators of the system studied, either by calculation or by estimation. We will present the Markov analysis of a system whose state space is finite as well as its estimator obtained using SPNs.